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# Group (mathematics)
## Elementary consequences of the group axioms {#elementary_consequences_of_the_group_axioms}
### Equivalent definition with relaxed axioms {#equivalent_definition_with_relaxed_axioms}
The group axioms for identity and inverses may be \"weakened\" to assert only the existence of a left identity and left inverses. From these *one-sided axioms*, one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are not weaker.
In particular, assuming associativity and the existence of a left identity $e$ (that is, `{{tmath|1= e\cdot f=f }}`{=mediawiki}) and a left inverse $f^{-1}$ for each element $f$ (that is, `{{tmath|1= f^{-1}\cdot f=e }}`{=mediawiki}), it follows that every left inverse is also a right inverse of the same element as follows. Indeed, one has
: \\begin{align}
f \\cdot f\^{-1}
` &=e \cdot (f \cdot f^{-1}) `\
` && \text{(left identity)}\\`\
` &=((f^{-1})^{-1} \cdot f^{-1}) \cdot (f \cdot f^{-1})`\
` && \text{(left inverse)}\\`\
` &=(f^{-1})^{-1} \cdot ((f^{-1} \cdot f) \cdot f^{-1})`\
` && \text{(associativity)}\\`\
` &=(f^{-1})^{-1} \cdot (e \cdot f^{-1})`\
` && \text{(left inverse)}\\`\
` &=(f^{-1})^{-1} \cdot f^{-1}`\
` && \text{(left identity)}\\`\
` &=e`\
` && \text{(left inverse)}`
\\end{align}
Similarly, the left identity is also a right identity:
: \\begin{align}
f\\cdot e
` &= f \cdot ( f^{-1} \cdot f)`\
` && \text{(left inverse)}\\`\
` &= (f \cdot f^{-1}) \cdot f`\
` && \text{(associativity)}\\`\
` &= e \cdot f`\
` && \text{(right inverse)}\\`\
` &= f`\
` && \text{(left identity)}`
\\end{align}
These results do not hold if any of these axioms (associativity, existence of left identity and existence of left inverse) is removed. For a structure with a looser definition (like a semigroup) one may have, for example, that a left identity is not necessarily a right identity.
The same result can be obtained by only assuming the existence of a right identity and a right inverse.
However, only assuming the existence of a *left* identity and a *right* inverse (or vice versa) is not sufficient to define a group. For example, consider the set $G = \{ e,f \}$ with the operator $\cdot$ satisfying $e \cdot e = f \cdot e = e$ and `{{tmath|1= e \cdot f = f \cdot f = f }}`{=mediawiki}. This structure does have a left identity (namely, `{{tmath|1= e }}`{=mediawiki}), and each element has a right inverse (which is $e$ for both elements). Furthermore, this operation is associative (since the product of any number of elements is always equal to the rightmost element in that product, regardless of the order in which these operations are applied). However, $( G , \cdot )$ is not a group, since it lacks a right identity.
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# Group (mathematics)
## Basic concepts {#basic_concepts}
When studying sets, one uses concepts such as subset, function, and quotient by an equivalence relation. When studying groups, one uses instead subgroups, homomorphisms, and quotient groups. These are the analogues that take the group structure into account.
### Group homomorphisms {#group_homomorphisms}
Group homomorphisms are functions that respect group structure; they may be used to relate two groups. A *homomorphism* from a group $(G,\cdot)$ to a group $(H,*)$ is a function $\varphi : G\to H$ such that `{{Block indent|left=1.6|<math>\varphi(a\cdot b)=\varphi(a)*\varphi(b)</math> for all elements <math>a</math> and <math>b</math> in {{tmath|1= G }}.}}`{=mediawiki} It would be natural to require also that $\varphi$ respect identities, `{{tmath|1= \varphi(1_G)=1_H }}`{=mediawiki}, and inverses, $\varphi(a^{-1})=\varphi(a)^{-1}$ for all $a$ in `{{tmath|1= G }}`{=mediawiki}. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.
The *identity homomorphism* of a group $G$ is the homomorphism $\iota_G : G\to G$ that maps each element of $G$ to itself. An *inverse homomorphism* of a homomorphism $\varphi : G\to H$ is a homomorphism $\psi : H\to G$ such that $\psi\circ\varphi=\iota_G$ and `{{tmath|1= \varphi\circ\psi=\iota_H }}`{=mediawiki}, that is, such that $\psi\bigl(\varphi(g)\bigr)=g$ for all $g$ in $G$ and such that $\varphi\bigl(\psi(h)\bigr)=h$ for all $h$ in `{{tmath|1= H }}`{=mediawiki}. An *isomorphism* is a homomorphism that has an inverse homomorphism; equivalently, it is a bijective homomorphism. Groups $G$ and $H$ are called *isomorphic* if there exists an isomorphism `{{tmath|1= \varphi : G\to H }}`{=mediawiki}. In this case, $H$ can be obtained from $G$ simply by renaming its elements according to the function `{{tmath|1= \varphi }}`{=mediawiki}; then any statement true for $G$ is true for `{{tmath|1= H }}`{=mediawiki}, provided that any specific elements mentioned in the statement are also renamed.
The collection of all groups, together with the homomorphisms between them, form a category, the category of groups.
An injective homomorphism $\phi : G' \to G$ factors canonically as an isomorphism followed by an inclusion, $G' \;\stackrel{\sim}{\to}\; H \hookrightarrow G$ for some subgroup `{{tmath|1= H }}`{=mediawiki} of `{{tmath|1= G }}`{=mediawiki}. Injective homomorphisms are the monomorphisms in the category of groups.
### Subgroups
Informally, a *subgroup* is a group $H$ contained within a bigger one, `{{tmath|1= G }}`{=mediawiki}: it has a subset of the elements of `{{tmath|1= G }}`{=mediawiki}, with the same operation. Concretely, this means that the identity element of $G$ must be contained in `{{tmath|1= H }}`{=mediawiki}, and whenever $h_1$ and $h_2$ are both in `{{tmath|1= H }}`{=mediawiki}, then so are $h_1\cdot h_2$ and `{{tmath|1= h_1^{-1} }}`{=mediawiki}, so the elements of `{{tmath|1= H }}`{=mediawiki}, equipped with the group operation on $G$ restricted to `{{tmath|1= H }}`{=mediawiki}, indeed form a group. In this case, the inclusion map $H \to G$ is a homomorphism.
In the example of symmetries of a square, the identity and the rotations constitute a subgroup `{{tmath|1= R=\{\mathrm{id},r_1,r_2,r_3\} }}`{=mediawiki}, highlighted in red in the Cayley table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. The subgroup test provides a necessary and sufficient condition for a nonempty subset `{{tmath|1= H }}`{=mediawiki} of a group `{{tmath|1= G }}`{=mediawiki} to be a subgroup: it is sufficient to check that $g^{-1}\cdot h\in H$ for all elements $g$ and $h$ in `{{tmath|1= H }}`{=mediawiki}. Knowing a group\'s subgroups is important in understanding the group as a whole.
Given any subset $S$ of a group `{{tmath|1= G }}`{=mediawiki}, the subgroup generated by $S$ consists of all products of elements of $S$ and their inverses. It is the smallest subgroup of $G$ containing `{{tmath|1= S }}`{=mediawiki}. In the example of symmetries of a square, the subgroup generated by $r_2$ and $f_{\mathrm{v}}$ consists of these two elements, the identity element `{{tmath|1= \mathrm{id} }}`{=mediawiki}, and the element `{{tmath|1= f_{\mathrm{h} }=f_{\mathrm{v} }\cdot r_2 }}`{=mediawiki}. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.
### Cosets
In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup $H$ determines left and right cosets, which can be thought of as translations of $H$ by an arbitrary group element `{{tmath|1= g }}`{=mediawiki}. In symbolic terms, the *left* and *right* cosets of `{{tmath|1= H }}`{=mediawiki}, containing an element `{{tmath|1= g }}`{=mediawiki}, are
The left cosets of any subgroup $H$ form a partition of `{{tmath|1= G }}`{=mediawiki}; that is, the union of all left cosets is equal to $G$ and two left cosets are either equal or have an empty intersection. The first case $g_1H=g_2H$ happens precisely when `{{tmath|1= g_1^{-1}\cdot g_2\in H }}`{=mediawiki}, i.e., when the two elements differ by an element of `{{tmath|1= H }}`{=mediawiki}. Similar considerations apply to the right cosets of `{{tmath|1= H }}`{=mediawiki}. The left cosets of $H$ may or may not be the same as its right cosets. If they are (that is, if all $g$ in $G$ satisfy `{{tmath|1= gH=Hg }}`{=mediawiki}), then $H$ is said to be a *normal subgroup*.
In `{{tmath|1= \mathrm{D}_4 }}`{=mediawiki}, the group of symmetries of a square, with its subgroup $R$ of rotations, the left cosets $gR$ are either equal to `{{tmath|1= R }}`{=mediawiki}, if $g$ is an element of $R$ itself, or otherwise equal to $U=f_{\mathrm{c}}R=\{f_{\mathrm{c}},f_{\mathrm{d}},f_{\mathrm{v}},f_{\mathrm{h}}\}$ (highlighted in green in the Cayley table of `{{tmath|1= \mathrm{D}_4 }}`{=mediawiki}). The subgroup $R$ is normal, because $f_{\mathrm{c}}R=U=Rf_{\mathrm{c}}$ and similarly for the other elements of the group. (In fact, in the case of `{{tmath|1= \mathrm{D}_4 }}`{=mediawiki}, the cosets generated by reflections are all equal: `{{tmath|1= f_{\mathrm{h} }R=f_{\mathrm{v} }R=f_{\mathrm{d} }R=f_{\mathrm{c} }R }}`{=mediawiki}.)
### Quotient groups {#quotient_groups}
Suppose that $N$ is a normal subgroup of a group `{{tmath|1= G }}`{=mediawiki}, and $G/N = \{gN \mid g\in G\}$ denotes its set of cosets. Then there is a unique group law on $G/N$ for which the map $G\to G/N$ sending each element $g$ to $gN$ is a homomorphism. Explicitly, the product of two cosets $gN$ and $hN$ is `{{tmath|1= (gh)N }}`{=mediawiki}, the coset $eN = N$ serves as the identity of `{{tmath|1= G/N }}`{=mediawiki}, and the inverse of $gN$ in the quotient group is `{{tmath|1= (gN)^{-1} = \left(g^{-1}\right)N }}`{=mediawiki}. The group `{{tmath|1= G/N }}`{=mediawiki}, read as \"`{{tmath|1= G }}`{=mediawiki} modulo `{{tmath|1= N }}`{=mediawiki}\", is called a *quotient group* or *factor group*. The quotient group can alternatively be characterized by a universal property.
$\cdot$ $R$ $U$
--------- ----- -----
$R$ $R$ $U$
$U$ $U$ $R$
: Cayley table of the quotient group $\mathrm{D}_4/R$
The elements of the quotient group $\mathrm{D}_4/R$ are $R$ and `{{tmath|1= U=f_{\mathrm{v} }R }}`{=mediawiki}. The group operation on the quotient is shown in the table. For example, `{{tmath|1= U\cdot U=f_{\mathrm{v} }R\cdot f_{\mathrm{v} }R=(f_{\mathrm{v} }\cdot f_{\mathrm{v} })R=R }}`{=mediawiki}. Both the subgroup $R=\{\mathrm{id},r_1,r_2,r_3\}$ and the quotient $\mathrm{D}_4/R$ are abelian, but $\mathrm{D}_4$ is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the semidirect product construction; $\mathrm{D}_4$ is an example.
The first isomorphism theorem implies that any surjective homomorphism $\phi : G \to H$ factors canonically as a quotient homomorphism followed by an isomorphism: `{{tmath|1= G \to G/\ker \phi \;\stackrel{\sim}{\to}\; H }}`{=mediawiki}. Surjective homomorphisms are the epimorphisms in the category of groups.
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# Group (mathematics)
## Basic concepts {#basic_concepts}
### Presentations
Every group is isomorphic to a quotient of a free group, in many ways.
For example, the dihedral group $\mathrm{D}_4$ is generated by the right rotation $r_1$ and the reflection $f_{\mathrm{v}}$ in a vertical line (every element of $\mathrm{D}_4$ is a finite product of copies of these and their inverses). Hence there is a surjective homomorphism `{{tmath|1= \phi }}`{=mediawiki} from the free group $\langle r,f \rangle$ on two generators to $\mathrm{D}_4$ sending $r$ to $r_1$ and $f$ to `{{tmath|1= f_1 }}`{=mediawiki}. Elements in $\ker \phi$ are called *relations*; examples include `{{tmath|1= r^4,f^2,(r \cdot f)^2 }}`{=mediawiki}. In fact, it turns out that $\ker \phi$ is the smallest normal subgroup of $\langle r,f \rangle$ containing these three elements; in other words, all relations are consequences of these three. The quotient of the free group by this normal subgroup is denoted `{{tmath|1= \langle r,f \mid r^4=f^2=(r\cdot f)^2=1 \rangle }}`{=mediawiki}. This is called a *presentation* of $\mathrm{D}_4$ by generators and relations, because the first isomorphism theorem for `{{tmath|1= \phi }}`{=mediawiki} yields an isomorphism `{{tmath|1= \langle r,f \mid r^4=f^2=(r\cdot f)^2=1 \rangle \to \mathrm{D}_4 }}`{=mediawiki}.
A presentation of a group can be used to construct the Cayley graph, a graphical depiction of a discrete group.
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# Group (mathematics)
## Examples and applications {#examples_and_applications}
thumb\|right\|A periodic wallpaper pattern gives rise to a wallpaper group. Examples and applications of groups abound. A starting point is the group $\Z$ of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra.
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.
thumb\|right\|The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers under addition. Elements of the fundamental group of a topological space are equivalence classes of loops, where loops are considered equivalent if one can be smoothly deformed into another, and the group operation is \"concatenation\" (tracing one loop then the other). For example, as shown in the figure, if the topological space is the plane with one point removed, then loops which do not wrap around the missing point (blue) can be smoothly contracted to a single point and are the identity element of the fundamental group. A loop which wraps around the missing point $k$ times cannot be deformed into a loop which wraps $m$ times (with `{{tmath|1= m\neq k }}`{=mediawiki}), because the loop cannot be smoothly deformed across the hole, so each class of loops is characterized by its winding number around the missing point. The resulting group is isomorphic to the integers under addition.
In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups. Further branches crucially applying groups include algebraic geometry and number theory.
In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory, in particular when implemented for finite groups. Applications of group theory are not restricted to mathematics; sciences such as physics, chemistry and computer science benefit from the concept.
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# Group (mathematics)
## Examples and applications {#examples_and_applications}
### Numbers
Many number systems, such as the integers and the rationals, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups.
#### Integers
The group of integers $\Z$ under addition, denoted `{{tmath|1= \left(\Z,+\right) }}`{=mediawiki}, has been described above. The integers, with the operation of multiplication instead of addition, $\left(\Z,\cdot\right)$ do *not* form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, $a=2$ is an integer, but the only solution to the equation $a\cdot b=1$ in this case is `{{tmath|1= b=\tfrac{1}{2} }}`{=mediawiki}, which is a rational number, but not an integer. Hence not every element of $\Z$ has a (multiplicative) inverse.
#### Rationals
The desire for the existence of multiplicative inverses suggests considering fractions $$\\frac{a}{b}.
Fractions of integers (with $b$ nonzero) are known as rational numbers. The set of all such irreducible fractions is commonly denoted `{{tmath|1= \Q }}`{=mediawiki}. There is still a minor obstacle for `{{tmath|1= \left(\Q,\cdot\right) }}`{=mediawiki}, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no $x$ such that `{{tmath|1= x\cdot 0=1 }}`{=mediawiki}), $\left(\Q,\cdot\right)$ is still not a group.
However, the set of all *nonzero* rational numbers $\Q\smallsetminus\left\{0\right\}=\left\{q\in\Q\mid q\neq 0\right\}$ does form an abelian group under multiplication, also denoted `{{tmath|1= \Q^{\times} }}`{=mediawiki}. Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of $a/b$ is `{{tmath|1= b/a }}`{=mediawiki}, therefore the axiom of the inverse element is satisfied.
The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and -- if division by other than zero is possible, such as in $\Q$ -- fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.
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# Group (mathematics)
## Examples and applications {#examples_and_applications}
### Modular arithmetic {#modular_arithmetic}
Modular arithmetic for a *modulus* $n$ defines any two elements $a$ and $b$ that differ by a multiple of $n$ to be equivalent, denoted by `{{tmath|1= a \equiv b\pmod{n} }}`{=mediawiki}. Every integer is equivalent to one of the integers from $0$ to `{{tmath|1= n-1 }}`{=mediawiki}, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent representative. Modular addition, defined in this way for the integers from $0$ to `{{tmath|1= n-1 }}`{=mediawiki}, forms a group, denoted as $\mathrm{Z}_n$ or `{{tmath|1= (\Z/n\Z,+) }}`{=mediawiki}, with $0$ as the identity element and $n-a$ as the inverse element of `{{tmath|1= a }}`{=mediawiki}.
A familiar example is addition of hours on the face of a clock, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on $9$ and is advanced $4$ hours, it ends up on `{{tmath|1= 1 }}`{=mediawiki}, as shown in the illustration. This is expressed by saying that $9+4$ is congruent to $1$ \"modulo `{{tmath|1= 12 }}`{=mediawiki}\" or, in symbols, $9+4\equiv 1 \pmod{12}.$
For any prime number `{{tmath|1= p }}`{=mediawiki}, there is also the multiplicative group of integers modulo `{{tmath|1= p }}`{=mediawiki}. Its elements can be represented by $1$ to `{{tmath|1= p-1 }}`{=mediawiki}. The group operation, multiplication modulo `{{tmath|1= p }}`{=mediawiki}, replaces the usual product by its representative, the remainder of division by `{{tmath|1= p }}`{=mediawiki}. For example, for `{{tmath|1= p=5 }}`{=mediawiki}, the four group elements can be represented by `{{tmath|1= 1,2,3,4 }}`{=mediawiki}. In this group, `{{tmath|1= 4\cdot 4\equiv 1\bmod 5 }}`{=mediawiki}, because the usual product $16$ is equivalent to `{{tmath|1= 1 }}`{=mediawiki}: when divided by $5$ it yields a remainder of `{{tmath|1= 1 }}`{=mediawiki}. The primality of $p$ ensures that the usual product of two representatives is not divisible by `{{tmath|1= p }}`{=mediawiki}, and therefore that the modular product is nonzero. The identity element is represented by `{{tmath|1= 1 }}`{=mediawiki}, and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer $a$ not divisible by `{{tmath|1= p }}`{=mediawiki}, there exists an integer $b$ such that $a\cdot b\equiv 1\pmod{p},$ that is, such that $p$ evenly divides `{{tmath|1= a\cdot b-1 }}`{=mediawiki}. The inverse $b$ can be found by using Bézout\'s identity and the fact that the greatest common divisor $\gcd(a,p)$ equals `{{tmath|1= 1 }}`{=mediawiki}. In the case $p=5$ above, the inverse of the element represented by $4$ is that represented by `{{tmath|1= 4 }}`{=mediawiki}, and the inverse of the element represented by $3$ is represented by `{{tmath|1= 2 }}`{=mediawiki}, as `{{tmath|1= 3\cdot 2=6\equiv 1\bmod{5} }}`{=mediawiki}. Hence all group axioms are fulfilled. This example is similar to $\left(\Q\smallsetminus\left\{0\right\},\cdot\right)$ above: it consists of exactly those elements in the ring $\Z/p\Z$ that have a multiplicative inverse. These groups, denoted `{{tmath|1= \mathbb F_p^\times }}`{=mediawiki}, are crucial to public-key cryptography.
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# Group (mathematics)
## Examples and applications {#examples_and_applications}
### Cyclic groups {#cyclic_groups}
A *cyclic group* is a group all of whose elements are powers of a particular element `{{tmath|1= a }}`{=mediawiki}. In multiplicative notation, the elements of the group are $\dots, a^{-3}, a^{-2}, a^{-1}, a^0, a, a^2, a^3, \dots,$ where $a^2$ means `{{tmath|1= a\cdot a }}`{=mediawiki}, $a^{-3}$ stands for `{{tmath|1= a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a)^{-1} }}`{=mediawiki}, etc. Such an element $a$ is called a generator or a primitive element of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as $\dots, (-a)+(-a), -a, 0, a, a+a, \dots.$
In the groups $(\Z/n\Z,+)$ introduced above, the element $1$ is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are `{{tmath|1= 1 }}`{=mediawiki}. Any cyclic group with $n$ elements is isomorphic to this group. A second example for cyclic groups is the group of `{{tmath|1= n }}`{=mediawiki}th complex roots of unity, given by complex numbers $z$ satisfying `{{tmath|1= z^n=1 }}`{=mediawiki}. These numbers can be visualized as the vertices on a regular $n$-gon, as shown in blue in the image for `{{tmath|1= n=6 }}`{=mediawiki}. The group operation is multiplication of complex numbers. In the picture, multiplying with $z$ corresponds to a counter-clockwise rotation by 60°. From field theory, the group $\mathbb F_p^\times$ is cyclic for prime $p$: for example, if `{{tmath|1= p=5 }}`{=mediawiki}, $3$ is a generator since `{{tmath|1= 3^1=3 }}`{=mediawiki}, `{{tmath|1= 3^2=9\equiv 4 }}`{=mediawiki}, `{{tmath|1= 3^3\equiv 2 }}`{=mediawiki}, and `{{tmath|1= 3^4\equiv 1 }}`{=mediawiki}.
Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element `{{tmath|1= a }}`{=mediawiki}, all the powers of $a$ are distinct; despite the name \"cyclic group\", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to `{{tmath|1= (\Z, +) }}`{=mediawiki}, the group of integers under addition introduced above. As these two prototypes are both abelian, so are all cyclic groups.
The study of finitely generated abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian.
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# Group (mathematics)
## Examples and applications {#examples_and_applications}
### Symmetry groups {#symmetry_groups}
upright=.75\|thumb\|The (2,3,7) triangle group, a hyperbolic reflection group, acts on this tiling of the hyperbolic plane
*Symmetry groups* are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below). Conceptually, group theory can be thought of as the study of symmetry. Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object `{{tmath|1= X }}`{=mediawiki} if every group element can be associated to some operation on `{{tmath|1= X }}`{=mediawiki} and the composition of these operations follows the group law. For example, an element of the (2,3,7) triangle group acts on a triangular tiling of the hyperbolic plane by permuting the triangles. By a group action, the group pattern is connected to the structure of the object being acted on.
In chemistry, point groups describe molecular symmetries, while space groups describe crystal symmetries in crystallography. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties. For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.
Group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.
Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons.
+-----------------------------------------------------------------------+--------------------------------------------------------------------------------------------------+-------------------------------------------------------------------+-----------------------------------------------------------------------------------+
| | | | |
+-----------------------------------------------------------------------+--------------------------------------------------------------------------------------------------+-------------------------------------------------------------------+-----------------------------------------------------------------------------------+
| Buckminsterfullerene displays`{{br}}`{=mediawiki}icosahedral symmetry | Ammonia, NH~3~. Its symmetry group is of order 6, generated by a 120° rotation and a reflection. | Cubane C~8~H~8~ features`{{br}}`{=mediawiki} octahedral symmetry. | The tetrachloroplatinate(II) ion, \[PtCl~4~\]^2−^ exhibits square-planar geometry |
+-----------------------------------------------------------------------+--------------------------------------------------------------------------------------------------+-------------------------------------------------------------------+-----------------------------------------------------------------------------------+
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved. Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
### General linear group and representation theory {#general_linear_group_and_representation_theory}
Matrix groups consist of matrices together with matrix multiplication. The *general linear group* $\mathrm {GL}(n, \R)$ consists of all invertible `{{tmath|1= n }}`{=mediawiki}-by-`{{tmath|1= n }}`{=mediawiki} matrices with real entries. Its subgroups are referred to as *matrix groups* or *linear groups*. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group `{{tmath|1= \mathrm{SO}(n) }}`{=mediawiki}. It describes all possible rotations in $n$ dimensions. Rotation matrices in this group are used in computer graphics.
*Representation theory* is both an application of the group concept and important for a deeper understanding of groups. It studies the group by its group actions on other spaces. A broad class of group representations are linear representations in which the group acts on a vector space, such as the three-dimensional Euclidean space `{{tmath|1= \R^3 }}`{=mediawiki}. A representation of a group $G$ on an $n$-dimensional real vector space is simply a group homomorphism $\rho : G \to \mathrm {GL}(n, \R)$ from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.
A group action gives further means to study the object being acted on. On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups.
### Galois groups {#galois_groups}
*Galois groups* were developed to help solve polynomial equations by capturing their symmetry features. For example, the solutions of the quadratic equation $ax^2+bx+c=0$ are given by $x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}.$ Each solution can be obtained by replacing the $\pm$ sign by $+$ or `{{tmath|1= - }}`{=mediawiki}; analogous formulae are known for cubic and quartic equations, but do *not* exist in general for degree 5 and higher. In the quadratic formula, changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomial equations and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular their solvability) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and roots similar to the formula above.
Modern Galois theory generalizes the above type of Galois groups by shifting to field theory and considering field extensions formed as the splitting field of a polynomial. This theory establishes---via the fundamental theorem of Galois theory---a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.
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# Group (mathematics)
## Finite groups {#finite_groups}
A group is called *finite* if it has a finite number of elements. The number of elements is called the order of the group. An important class is the *symmetric groups* `{{tmath|1= \mathrm{S}_N }}`{=mediawiki}, the groups of permutations of $N$ objects. For example, the symmetric group on 3 letters $\mathrm{S}_3$ is the group of all possible reorderings of the objects. The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 (factorial of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group $\mathrm{S}_N$ for a suitable integer `{{tmath|1= N }}`{=mediawiki}, according to Cayley\'s theorem. Parallel to the group of symmetries of the square above, $\mathrm{S}_3$ can also be interpreted as the group of symmetries of an equilateral triangle.
The order of an element $a$ in a group $G$ is the least positive integer $n$ such that `{{tmath|1= a^n=e }}`{=mediawiki}, where $a^n$ represents $\underbrace{a \cdots a}_{n \text{ factors}},$ that is, application of the operation \"`{{tmath|1= \cdot }}`{=mediawiki}\" to $n$ copies of `{{tmath|1= a }}`{=mediawiki}. (If \"`{{tmath|1= \cdot }}`{=mediawiki}\" represents multiplication, then $a^n$ corresponds to the `{{tmath|1= n }}`{=mediawiki}th power of `{{tmath|1= a }}`{=mediawiki}.) In infinite groups, such an $n$ may not exist, in which case the order of $a$ is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element.
More sophisticated counting techniques, for example, counting cosets, yield more precise statements about finite groups: Lagrange\'s Theorem states that for a finite group $G$ the order of any finite subgroup $H$ divides the order of `{{tmath|1= G }}`{=mediawiki}. The Sylow theorems give a partial converse.
The dihedral group $\mathrm{D}_4$ of symmetries of a square is a finite group of order 8. In this group, the order of $r_1$ is 4, as is the order of the subgroup $R$ that this element generates. The order of the reflection elements $f_{\mathrm{v}}$ etc. is 2. Both orders divide 8, as predicted by Lagrange\'s theorem. The groups $\mathbb F_p^\times$ of multiplication modulo a prime $p$ have order `{{tmath|1= p-1 }}`{=mediawiki}.
### Finite abelian groups {#finite_abelian_groups}
Any finite abelian group is isomorphic to a product of finite cyclic groups; this statement is part of the fundamental theorem of finitely generated abelian groups.
Any group of prime order $p$ is isomorphic to the cyclic group $\mathrm{Z}_p$ (a consequence of Lagrange\'s theorem). Any group of order $p^2$ is abelian, isomorphic to $\mathrm{Z}_{p^2}$ or `{{tmath|1= \mathrm{Z}_p \times \mathrm{Z}_p }}`{=mediawiki}. But there exist nonabelian groups of order `{{tmath|1= p^3 }}`{=mediawiki}; the dihedral group $\mathrm{D}_4$ of order $2^3$ above is an example.
### Simple groups {#simple_groups}
When a group $G$ has a normal subgroup $N$ other than $\{1\}$ and $G$ itself, questions about $G$ can sometimes be reduced to questions about $N$ and `{{tmath|1= G/N }}`{=mediawiki}. A nontrivial group is called *simple* if it has no such normal subgroup. Finite simple groups are to finite groups as prime numbers are to positive integers: they serve as building blocks, in a sense made precise by the Jordan--Hölder theorem.
### Classification of finite simple groups {#classification_of_finite_simple_groups}
Computer algebra systems have been used to list all groups of order up to 2000. But classifying all finite groups is a problem considered too hard to be solved.
The classification of all finite *simple* groups was a major achievement in contemporary group theory. There are several infinite families of such groups, as well as 26 \"sporadic groups\" that do not belong to any of the families. The largest sporadic group is called the monster group. The monstrous moonshine conjectures, proved by Richard Borcherds, relate the monster group to certain modular functions.
The gap between the classification of simple groups and the classification of all groups lies in the extension problem.
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# Group (mathematics)
## Groups with additional structure {#groups_with_additional_structure}
An equivalent definition of group consists of replacing the \"there exist\" part of the group axioms by operations whose result is the element that must exist. So, a group is a set $G$ equipped with a binary operation $G \times G \rightarrow G$ (the group operation), a unary operation $G \rightarrow G$ (which provides the inverse) and a nullary operation, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoids existential quantifiers and is used in computing with groups and for computer-aided proofs.
This way of defining groups lends itself to generalizations such as the notion of group object in a category. Briefly, this is an object with morphisms that mimic the group axioms.
### Topological groups {#topological_groups}
*Main article: Topological group* Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally, $g \cdot h$ and $g^{-1}$ must not vary wildly if $g$ and $h$ vary only a little. Such groups are called *topological groups,* and they are the group objects in the category of topological spaces. The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any other topological field, such as the field of complex numbers or the field of `{{math|''p''}}`{=mediawiki}-adic numbers . These examples are locally compact, so they have Haar measures and can be studied via harmonic analysis. Other locally compact topological groups include the group of points of an algebraic group over a local field or adele ring; these are basic to number theory Galois groups of infinite algebraic field extensions are equipped with the Krull topology, which plays a role in infinite Galois theory. A generalization used in algebraic geometry is the étale fundamental group.
### Lie groups {#lie_groups}
A *Lie group* is a group that also has the structure of a differentiable manifold; informally, this means that it looks locally like a Euclidean space of some fixed dimension. Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be smooth.
A standard example is the general linear group introduced above: it is an open subset of the space of all $n$-by-$n$ matrices, because it is given by the inequality $\det (A) \ne 0,$ where $A$ denotes an $n$-by-$n$ matrix.
Lie groups are of fundamental importance in modern physics: Noether\'s theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time, are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models---imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. Another example is the group of Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves---in the absence of significant gravitation---as a model of spacetime in special relativity. The full symmetry group of Minkowski space, i.e., including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory. An important example of a gauge theory is the Standard Model, which describes three of the four known fundamental forces and classifies all known elementary particles.
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# Group (mathematics)
## Generalizations
More general structures may be defined by relaxing some of the axioms defining a group. The table gives a list of several structures generalizing groups.
For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers $\mathbb N$ (including zero) under addition form a monoid, as do the nonzero integers under multiplication `{{tmath|1= (\Z \smallsetminus \{0\}, \cdot) }}`{=mediawiki}. Adjoining inverses of all elements of the monoid $(\Z \smallsetminus \{0\}, \cdot)$ produces a group `{{tmath|1= (\Q \smallsetminus \{0 \}, \cdot) }}`{=mediawiki}, and likewise adjoining inverses to any (abelian) monoid `{{tmath|1= M }}`{=mediawiki} produces a group known as the Grothendieck group of `{{tmath|1= M }}`{=mediawiki}.
A group can be thought of as a small category with one object `{{tmath|1= x }}`{=mediawiki} in which every morphism is an isomorphism: given such a category, the set $\operatorname{Hom}(x,x)$ is a group; conversely, given a group `{{tmath|1= G }}`{=mediawiki}, one can build a small category with one object `{{tmath|1= x }}`{=mediawiki} in which `{{tmath|1= \operatorname{Hom}(x,x) \simeq G }}`{=mediawiki}. More generally, a groupoid is any small category in which every morphism is an isomorphism. In a groupoid, the set of all morphisms in the category is usually not a group, because the composition is only partially defined: `{{tmath|1= fg }}`{=mediawiki} is defined only when the source of `{{tmath|1= f }}`{=mediawiki} matches the target of `{{tmath|1= g }}`{=mediawiki}. Groupoids arise in topology (for instance, the fundamental groupoid) and in the theory of stacks.
Finally, it is possible to generalize any of these concepts by replacing the binary operation with an `{{mvar|n}}`{=mediawiki}-ary operation (i.e., an operation taking `{{mvar|n}}`{=mediawiki} arguments, for some nonnegative integer `{{mvar|n}}`{=mediawiki}). With the proper generalization of the group axioms, this gives a notion of `{{mvar|n}}`{=mediawiki}-ary group
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# Max Horkheimer
**Max Horkheimer** (`{{IPAc-en|ˈ|h|ɔːr|k|h|aɪ|m|ər}}`{=mediawiki} `{{respell|HORK|hy|mər}}`{=mediawiki}; `{{IPA|de|ˈhɔɐ̯kˌhaɪmɐ|lang}}`{=mediawiki}; 14 February 1895 -- 7 July 1973) was a German philosopher and sociologist best known for his role in developing critical theory as director of the Institute for Social Research, commonly associated with the Frankfurt School.
Advancing a materialist theory of reason and society, Horkheimer analyzed the rise of instrumental reason, the erosion of the concept of truth, the decline of individual autonomy, the social-psychological roots of authoritarianism, and the reproduction of domination under modern capitalism. These concerns became fundamental to critical theory.
His most influential works include *Eclipse of Reason* (1947), *Dialectic of Enlightenment* (1947, with Theodor W. Adorno), and a series of foundational essays written in the 1930s for the *Zeitschrift für Sozialforschung*, later collected in *Between Philosophy and Social Science* and *Critical Theory: Selected Essays*. He also composed aphoristic reflections between the late 1920s and the 1960s, published posthumously as *Dämmerung* (*Dawn and Decline*). As director of the Institute, Horkheimer planned, supported, and made possible many other significant works.
## Biography
### Early life {#early_life}
On 14 February 1895, Horkheimer was born the only son of Moritz and Babetta Horkheimer. Horkheimer was born into a conservative, wealthy Orthodox Jewish family. His father was a successful businessman who owned several textile factories in the Zuffenhausen district of Stuttgart, where Max was born. Moritz expected his son to follow in his footsteps and own the family business.
Max was taken out of school in 1910 to work in the family business, where he eventually became a junior manager. During this period he would begin two relationships that would last for the rest of his life. First, he met Friedrich Pollock, who would later become a close academic colleague, and who would remain Max\'s closest friend. He also met Rose Riekher, his father\'s personal secretary. Eight years Max\'s senior, a Christian, and from a lower economic class, Riekher was not considered a suitable match by Moritz Horkheimer. Despite this, Max and Maidon would marry in 1926 and remain together until her death in 1969.
In 1917, his manufacturing career ended and his chances of taking over his family business were interrupted when he was drafted into World War I. However, Horkheimer avoided service, being rejected on medical grounds.
### Education
In the spring of 1919, after failing an army physical, Horkheimer enrolled at Munich University. While living in Munich, he was mistaken for the revolutionary playwright Ernst Toller and arrested and imprisoned.
After being released, Horkheimer moved to Frankfurt am Main, where he studied philosophy and psychology under Hans Cornelius. There, he met Theodor Adorno, several years his junior, with whom he would strike a lasting friendship and a collaborative relationship. After an abortive attempt at writing a dissertation on Gestalt psychology, Horkheimer, with Cornelius\'s direction, completed his doctorate in philosophy with a 78-page dissertation titled *The Antinomy of Teleological Judgment* (*Zur Antinomie der teleologischen Urteilskraft*). In 1925, Horkheimer was habilitated with a dissertation entitled *Kant\'s Critique of Judgment as Mediation between Practical and Theoretical Philosophy* (*Über Kants Kritik der Urteilskraft als Bindeglied zwischen theoretischer und praktischer Philosophie*). There, he met Friedrich Pollock, who would be his colleague at the University of Frankfurt Institute for Social Research. The following year, Max was appointed *Privatdozent*. Shortly after, in 1926, Horkheimer married Rose Riekher.
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# Max Horkheimer
## Biography
### Institute of Social Research (Institut für Sozialforschung) {#institute_of_social_research_institut_für_sozialforschung}
In 1926, Horkheimer was an \"unsalaried lecturer in Frankfurt.\" Shortly after, in 1930, he was promoted to professor of philosophy at the University of Frankfurt am Main. In the same year, when the Institute for Social Research\'s directorship became vacant, after the departure of Carl Grünberg, Horkheimer was elected to the position \"by means of an endowment from a wealthy businessman\". The Institute had had its beginnings in a Marxist study group started by Felix Weil, a one-time student of political science at Frankfurt who used his inheritance to fund the group as a way to support his leftist academic aims. Pollock and Horkheimer were partners with Weil in the early activities of the institute.
Horkheimer worked to make the Institute a purely academic enterprise. As director, he changed the institute from an orthodox Marxist school to a heterodox school for critical social research. The following year publication of the institute\'s *Zeitschrift für Sozialforschung* began, with Horkheimer as its editor.
Horkheimer intellectually reoriented the institute, proposing a programme of collective research aimed at specific social groups (specifically the working class) that would highlight the problem of the relationship between history and reason. The Institute focused on integrating the views of Karl Marx and Sigmund Freud. The Frankfurt School attempted this by systematically hitching together the different conceptual structures of historical materialism and psychoanalysis.
During the time between Horkheimer\'s being named Professor of Social Philosophy and director of the Institute in 1930, the Nazi party became the second largest party in the Reichstag. In the midst of the violence surrounding the Nazis\' rise, Horkheimer and his associates began to prepare for the possibility of moving the Institute out of Germany. Horkheimer\'s *venia legendi* was revoked by the new Nazi government because of the Marxian nature of the institute\'s ideas as well as its prominent Jewish association. When Hitler was named the Chancellor in 1933, the institute was thus forced to close its location in Germany.
He emigrated to Geneva, Switzerland, and then to New York City the following year, where Horkheimer met with president of Columbia University Nicholas Murray Butler to discuss hosting the institute. To Horkheimer\'s surprise, the president agreed to host the Institute in exile as well as offer Horkheimer a building for the institute. In July 1934, Horkheimer accepted an offer from Columbia University to relocate the institute to one of their buildings.
In 1940, Horkheimer received American citizenship and moved to the Pacific Palisades district of Los Angeles, California, where his collaboration with Adorno would yield the *Dialectic of Enlightenment.* In 1942, Horkheimer assumed the directorship of the Scientific Division of the American Jewish Committee. In this capacity, he helped launch and organize a series of five Studies in Prejudice, which were published in 1949 and 1950. The most important of these was the pioneering study in social psychology entitled *The Authoritarian Personality*, itself a methodologically advanced reworking of some of the themes treated in a collective project produced by the Institute in its first years of exile, Studies in Authority and Family.
In the years that followed, Horkheimer did not publish much, although he continued to edit *Studies in Philosophy and Social Science* as a continuation of the *Zeitschrift.* In 1949, he returned to Frankfurt, where the Institute for Social Research reopened in 1950. Between 1951 and 1953 Horkheimer was rector of the University of Frankfurt am Main. In 1953, Horkheimer stepped down from director of the Institute and took on a smaller role in the institute, while Adorno became director.
### Later years {#later_years}
Horkheimer continued to teach at the university until his retirement in the mid-1960s. In 1953, he was awarded the Goethe Plaque of the City of Frankfurt, and was later named an honorary citizen of Frankfurt for life. He returned to the United States in 1954 and 1959 to lecture as a frequent visiting professor at the University of Chicago. In the late 1960s, Horkheimer supported Pope Paul VI\'s stand against artificial contraception, specifically the pill, arguing that it would lead to the end of romantic love.
### Legacy
He remained an important figure until his death in Nuremberg in 1973. Max Horkheimer with the help of Theodor Adorno, Herbert Marcuse, Walter Benjamin, Leo Löwenthal, Otto Kirchheimer, Frederick Pollock and Franz Neumann developed \"Critical Theory\". According to Larry Ray \"Critical Theory\" has \"become one of the most influential social theories of the twentieth century\".
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# Max Horkheimer
## Thought
Horkheimer\'s work is marked by a concern to show the relation between affect (especially suffering) and concepts (understood as action-guiding expressions of reason). In that, he responded critically to what he saw as the one-sidedness of both neo-Kantianism (with its focus on concepts) and *Lebensphilosophie* (with its focus on expression and world-disclosure). He did not think that either was wrong, but he insisted that the insights of each school on its own could not adequately contribute to the repair of social problems. Horkheimer focused on the connections between social structures, networks/subcultures and individual realities and concluded that we are affected and shaped by the proliferation of products on the marketplace. It is also important to note that Horkheimer collaborated with Herbert Marcuse, Erich Fromm, Theodor Adorno, and Walter Benjamin.
### Critical theory {#critical_theory}
Through critical theory, a social theory focusing on critiquing and changing society, Horkheimer \"attempted to revitalize radical social, and cultural criticism\" and discussed authoritarianism, militarism, economic disruption, environmental crisis and the poverty of mass culture. Horkheimer helped to create critical theory through a mix of radical and conservative lenses that stem from radical Marxism and end up in \"pessimistic Jewish transcendentalism\".
He developed his critical theory by examining his own wealth while witnessing the juxtaposition of the bourgeois and the impoverished. This critical theory embraced the future possibilities of society and was preoccupied with forces which moved society toward rational institutions that would ensure a true, free, and just life. He was convinced of the need to \"examine the entire material and spiritual culture of mankind\" in order to transform society as a whole. Horkheimer sought to enable the working class to reclaim their power in order to resist the lure of fascism. Horkheimer stated himself that \"the rationally organized society that regulates its own existence\" was necessary along with a society that could \"satisfy common needs\". To satisfy these needs, it reached out for a total understanding of history and knowledge. Through this, critical theory develops a \"critique of bourgeois society through which \'ideology critique\' attempted to locate the \'utopian content\' of dominant systems of thought\". Above all, critical theory sought to develop a critical perspective in the discussion of all social practices.
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# Max Horkheimer
## Thought
### Writing
#### *Between Philosophy and Social Science* {#between_philosophy_and_social_science}
*Between Philosophy and Social Science* appeared between 1930 and 1938, during the time the Frankfurt school moved from Frankfurt to Geneva to Columbia University. It included: \"Materialism and Morality\", \"The Present Situation of Moral Philosophy and the Tasks of an Institute for Social Research\", \"On the Problem of Truth\", \"Egoism and the Freedom Movement\", \"History and Psychology\", \"A New Concept of Ideology\", \"Remarks on Philosophical Anthropology\", and \"The Rationalism Debate in Contemporary Philosophy\". It also included \"The Present Situation of Social Philosophy and the Tasks for an Institute of Social Research\", \"Egoism and Freedom Movements\" and \"Beginnings of the Bourgeois Philosophy of History\". The essays within \"Between Philosophy and Social Science\" were Horkheimer\'s attempts to \"remove the individual from mass culture, a function for philosophy from the commodification of everything\". Horkheimer was extremely invested in the individual. In one of his writings, he states, \"When we speak of an individual as a historical entity, we mean not merely the space-time and the sense existence of a particular member of the human race, but in addition, his awareness of his own individuality as a conscious human being, including recognition of his own identity.\"
\"The Present Situation of Social Philosophy and the Tasks for an Institute of Social Research\" was not only included in this volume, but it was also used as Horkheimer\'s inaugural speech as director of the Frankfurt School. In this speech he related economic groups to the struggles and challenges of real life. Horkheimer often referenced human struggle and used this example in his speech because it was a topic he understood well.
\"Egoism and Freedom Movements\" and \"Beginnings of the Bourgeois Philosophy of History\" are the longest of the essays. The first is an evaluation of Machiavelli, Hobbes and Vico; the latter discusses the bourgeois control. In Beginnings of the Bourgeois Philosophy of History, Horkheimer explained \"what he learned from the bourgeois rise to power and what of the bourgeois he thought was worth preserving.
The volume also looks at the individual as the \"troubled center of philosophy.\" Horkheimer expressed that \"there is no formula that defines the relationship among individuals, society and nature for all time\". To understand the problem of the individual further, Horkheimer included two case studies on the individual: one on Montaigne and one on himself.
#### *Eclipse of Reason* {#eclipse_of_reason}
Horkheimer\'s book, *Eclipse of Reason*, started in 1941 and published in 1947, is broken into five sections: Means and Ends, Conflicting Panaceas, The Revolt of Nature, The Rise and Decline of the Individual, and On the Concept of Philosophy. The *Eclipse of Reason* focuses on the concept of reason within the history of Western philosophy, which can only be fostered in an environment of free, critical thinking while also linking positivist and instrumental reason with the rise of fascism. He distinguishes between objective, subjective and instrumental reason, and states that we have moved from the former through the center and into the latter (though subjective and instrumental reason are closely connected). Objective reason deals with universal truths that dictate that an action is either right or wrong. It is a concrete concept and a force in the world that requires specific modes of behavior. The focus in the objective faculty of reason is on the ends, rather than the means. Subjective reason is an abstract concept of reason, and focuses primarily on means. Specifically, the reasonable nature of the purpose of action is irrelevant -- the ends only serve the purpose of the subject (generally self-advancement or preservation). To be \"reasonable\" in this context is to be suited to a particular purpose, to be \"good for something else\". This aspect of reason is universally conforming, and easily furnishes ideology. In instrumental reason, the sole criterion of reason is its operational value or purposefulness, and with this, the idea of truth becomes contingent on mere subjective preference (hence the relation with subjective reason). Because subjective/instrumental reason rules, the ideals of a society, for example democratic ideals, become dependent on the \"interests\" of the people instead of being dependent on objective truths. Horkheimer writes, \"Social power is today more than ever mediated by power over things. The more intense an individual\'s concern with power over things, the more will things dominate him, the more will he lack any genuine individual traits, and the more will his mind be transformed into an automation of formalized reason.\"
Horkheimer acknowledges that objective reason has its roots in Reason (\"Logos\" in Greek) and concludes, \"If by enlightenment and intellectual progress we mean the freeing of man from superstitious belief in evil forces, in demons and fairies, in blind fate -- in short, the emancipation from fear -- then denunciation of what is currently called reason is the greatest service we can render.\"
#### *Dialectic of Enlightenment* {#dialectic_of_enlightenment}
Max Horkheimer and Theodor Adorno collaborated to publish *Dialectic of Enlightenment*, which was originally published in 1944. The inspiration for this piece came from when Horkheimer and Adorno had to flee Germany, because of Hitler, and go to New York. They went to America and \"absorbed the popular culture\"; thinking that it was a form of totalitarianism. Nonetheless, Dialectic of Enlightenment\'s main argument was to serve as a wide-ranging critique of the \"self-destruction of enlightenment\". The work criticized popular culture as \"the product of a culture industry whose goal was to stupefy the masses with endless mass produced copies of the same thing\" (Lemert). Along with that, Horkheimer and Adorno had a few arguments; one being that these mass-produced products only appear to change over time. Horkheimer and Adorno stated that these products were so standardized in order to help consumers comprehend and appreciate the products with little attention given to them. They expressed, \"the result is a constant reproduction of the same thing\" (Adorno and Horkheimer, 1993 \[1944\]). However, they also explain how pseudo-individuality is encouraged among these products in order to keep the consumers coming back for more. They argue that small differences in products within the same area are acceptable.
The similar patterns found in the content of popular culture (films, popular songs and radio) have the same central message; \"it\'s all linked to \"the necessity of obedience of the masses to the social hierarchy in place in advanced capitalist societies\". These products appeal to the masses and encourage conformity to the consumers. In return, capitalism remains in power while buyers continue to consume from the industry. This is dangerous because the consumers\' belief that the powers of technology are liberating, starts to increase. To support their claim, Horkheimer and Adorno, \"proposed an antidote: not just thinking the relations of *things*, but also, as an immediate second step, thinking *through* that thinking, self-reflexively.\" In other words, technology lacks self-reflexivity. Nonetheless, Horkheimer and Adorno believed that art was an exception, because it \"is an open-ended system with no fixed rules\"; thus, it could not be an object of the industry.
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# Max Horkheimer
## Criticisms
Perry Anderson sees Horkheimer\'s attempt to make the Institute purely academic as \"symptomatic of a more universal process, the emergence of a \'Western Marxism\' divorced from the working-class movement and dominated by academic philosophers and the \'product of defeat{{\'\"}} because of the isolation of the Russian Revolution. Rolf Wiggershaus, author of *The Frankfurt School* believed Horkheimer lacked the audacious theoretical construction produced by those like Marx and Lukács and that his main argument was that those living in misery had the right to material egoism. In his book, \"Social Theory\", Alex Callinicos claims that *Dialectic of Enlightenment* offers no systematic account of conception of rationality, but rather professes objective reason intransigently to an extent. Charles Lemert discusses in his book *Social Theory* that in writing *Dialectic of Enlightenment*, Horkheimer and Adorno lack sufficient sympathy for the cultural plight of the average working person, unfair to criticize the tastes of ordinary people, and that popular culture does not really buttress social conformity and stabilize capitalism as much as the Frankfurt school thinks.
Ingar Solty, in a February 2020 *Jacobin* magazine article, notes that the work of Horkheimer, Adorno, and the Frankfurt School as a whole is marked by \"the vast historical defeats suffered by the interwar socialist movement.\" He notes, \"Horkheimer and Adorno thus became increasingly pessimistic with regards to the working class\'s ability to overthrow capitalism \... Horkheimer did not conduct empirical research on capitalism and its crises \... the hierarchical nature of the international division of labor, the organization of internationalizing capitalism in a system of nation-states, the origins of imperialism and inter-imperial rivalries, or such \... For Horkheimer, the working class had been a revolutionary subject only in the abstract \... \[it\] was essentially an empty placeholder for the subject which would overthrow an economic and social system which they considered wrong. If it failed to live up to its expectations, then it could easily be replaced by another subject of revolution---or the conclusion that there was no way out (of capitalism).\"
Solty contextualizes Horkheimer\'s (and, by implication, the Frankfurt School\'s) \"return from \'revolutionary optimism\' to \'revolutionary pessimism{{\'\"}} by noting, \"\[m\]any postwar radical leftists and anti-capitalists, especially those not organized in real workers\' parties, were disappointed revolutionaries. The German writer Alfred Andersch, who had been close to the KPD before 1933 and then withdrawn into \"inner emigration,\" called the West German postwar left a \"homeless left.\" The working classes\' betrayals seemed to continue after 1945. After the short-lived socialist revival, the Cold War and the internationalization of the New Deal as the Keynesian welfare state seemed to have completely absorbed what was left of revolutionary working-class spirit. This led many disappointed leftists to culture and ideology as levels of analyses which could explain this failure of the working class.\" Solty identifies Horkheimer\'s (and, implicitly, the Frankfurt School\'s) work as an important influence on that of Michel Foucault:
> Ultimately, both Horkheimer and Foucault only considered the defense of remaining elements of freedom and the identification of \"micro-powers\" of domination a possibility, but changes in the macro-power structures were out of reach. In other words, a Left was born that was no longer oriented toward \"counter-hegemony\" (as per Antonio Gramsci), as a way of building toward power, but rather \"anti-hegemony\" (Horkheimer, Foucault, etc.), as John Sanbonmatsu put it in his critique of postmodernism.
## Selected works {#selected_works}
### Books
- *Authority and the Family* (1936)
- *Traditional and Critical Theory* (1937)
- *Dialectic of Enlightenment* (1947) -- with Theodor Adorno `{{ISBN|978-0-8264-0093-2}}`{=mediawiki}
- *Eclipse of Reason* (1947) (orig. 1941 \"The End of Reason\", *Studies in Philosophy and Social Sciences*, Vol. IX) `{{ISBN|978-1-4437-3041-9}}`{=mediawiki}
- *Egoism and the Freedom Movement*
- *The Longing for the Totally Other*
- *Critique of Instrumental Reason* (1967) `{{ISBN|978-0-8264-0088-8}}`{=mediawiki}
- *Critical Theory: Selected Essays* (1972) `{{ISBN|978-0-8264-0083-3}}`{=mediawiki}
- *Dawn & Decline* (1978) `{{ISBN|978-0-8164-9329-6}}`{=mediawiki}
- His collected works have been issued in German as *Max Horkheimer*: *Gesammelte Schriften* (1985--1996). 19 volumes, edited by Alfred Schmidt and Gunzelin Schmid Noerr. S. Fischer Verlag, Frankfurt am Main.
### Articles
- , in *Studies in Philosophy and Social Science*, vol. 8, n°3, New-York, 1939.
- \"The Authoritarian State\". 15 (Spring 1973). New York: Telos Press
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# Mervyn Peake
**Mervyn Laurence Peake** (9 July 1911 -- 17 November 1968) was a British writer, artist, poet, and illustrator. He is best known for what are usually referred to as the *Gormenghast* books. The four works were part of what Peake conceived as a lengthy cycle, the completion of which was prevented by his death. They are sometimes compared to the work of his older contemporary J. R. R. Tolkien, but Peake\'s surreal fiction was influenced by his early love for Charles Dickens and Robert Louis Stevenson rather than Tolkien\'s studies of mythology and philology.
Peake also wrote poetry and literary nonsense in verse form, short stories for adults and children (*Letters from a Lost Uncle*, 1948), stage and radio plays, and *Mr Pye* (1953), a relatively tightly structured novel in which God implicitly mocks the evangelical pretensions and cosy world-view of the eponymous hero.
Peake first made his reputation as a painter and illustrator during the 1930s and 1940s, when he lived in London, and he was commissioned to produce portraits of well-known people. For a short time at the end of World War II he was commissioned by various newspapers to depict war scenes. A collection of his drawings is still in the possession of his family. Although he gained little popular success in his lifetime, his work was highly respected by his peers, and his friends included C. S. Lewis, Dylan Thomas and Graham Greene. His works are now included in the collections of the National Portrait Gallery, the Imperial War Museum and The National Archives.
In 2008, *The Times* named Peake among their list of \"The 50 greatest British writers since 1945\".
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# Mervyn Peake
## Early life {#early_life}
Mervyn Peake was born of British parents in Kuling located on top of Mount Lu in Jiujiang in 1911, only three months before the revolution and the founding of the Republic of China. His father, Ernest Cromwell Peake, was a medical missionary doctor with the London Missionary Society of the Congregationalist tradition, and his mother, Amanda Elizabeth Powell, had come to China as a missionary assistant. Ernest and Amanda met in July 1903 at Kuling (from the English word \"cooling\"), a summer European missionary resort in Mount Lu about the Yangtze River in Jiujiang. They got married in Hong Kong in December of that same year.
The Peakes were given leave to visit England just before World War I in 1914 and returned to China in 1916. Mervyn Peake attended Tientsin Grammar School until the family left for England in December 1922 via the Trans-Siberian Railway. He would later write a novella about this time, titled *The White Chief of the Umzimbooboo Kaffirs*. Peake never returned to China but it has been noted that Chinese influences can be detected in his works, not least in the castle of Gormenghast itself, which in some respects echoes his birthplace Kuling, the ancient walled city of Beijing, as well as the enclosed compound where he grew up in Tianjin.`{{citation-needed|date=August 2024}}`{=mediawiki} It is also likely that his early exposure to the contrasts between the lives of the Europeans and of the Chinese, and between the poor and the wealthy in China, also exerted an influence on the Gormenghast books.`{{citation-needed|date=August 2024}}`{=mediawiki}
His education continued at Eltham College, Mottingham (1923--29), where his talents were encouraged by his English teacher, Eric Drake. Peake completed his formal education at Croydon School of Art in the autumn of 1929, and then from December 1929 to 1933 at the Royal Academy Schools, where he first painted in oils. By this time he had written his first long poem, *A Touch o\' the Ash*. In 1931, he had a painting accepted for display by the Royal Academy and exhibited his work with the so-called \"Soho Group\".
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# Mervyn Peake
## Career
His early career in the 1930s was as a painter in London, although he lived on the Channel Island of Sark for a time. He first moved to Sark in 1932 where his former teacher Eric Drake was setting up an artists\' colony. In 1934, Peake exhibited with the Sark artists both in the Sark Gallery built by Drake and at the Cooling Galleries in London, and in 1935 he exhibited at the Royal Academy and at the Leger Galleries in London.
In 1936, he returned to London and was commissioned to design the sets and costumes for *The Insect Play*, and his work was acclaimed in *The Sunday Times*. He also began teaching life drawing at Westminster School of Art where he met the painter Maeve Gilmore, whom he married in 1937. They had three children: Sebastian (1940--2012), Fabian (born 1942), and Clare (born 1949).
Peake had a very successful exhibition of paintings at the Calmann Gallery in London in 1938 and his first book, the self-illustrated children\'s pirate romance *Captain Slaughterboard Drops Anchor* (based on a story he had written around 1936), was first published in 1939 by *Country Life*. In December 1939, he was commissioned by Chatto & Windus to illustrate a children\'s book, *Ride a Cock Horse and Other Nursery Rhymes*, published for the Christmas market in 1940.
### Enlistment
At the outbreak of World War II, he applied to become a war artist, for he was keen to put his skills at the service of his country. He imagined *An Exhibition by the Artist, Adolf Hitler*, in which horrific images of war with ironic titles were offered as \"artworks\" by the Nazi leader. Although the drawings were bought by the British Ministry of Information, Peake\'s application was turned down and he was conscripted into the Army, where he served first with the Royal Artillery, then with the Royal Engineers. He began writing *Titus Groan* at this time.
In April 1942, after his requests for commissions as a war artist -- or even leave to depict war damage in London -- had been consistently refused, he suffered a nervous breakdown and was sent to Southport Hospital. That autumn he was taken on as a graphic artist by the Ministry of Information for a period of six months to work on propaganda illustrations. The next spring he was invalided out of the Army. In 1943 he was commissioned by the War Artists\' Advisory Committee, WAAC, to paint glassblowers at the Chance Brothers factory in Smethwick where cathode ray tubes for early radar sets were being produced. Peake was next given a full-time, three-month WAAC contract to depict various factory subjects and was also asked to submit a large painting showing RAF pilots being debriefed. Some of these paintings are on permanent display in Manchester Art Gallery whilst other examples are in the Imperial War Museum collection.
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# Mervyn Peake
## Career
### Illustration and writing {#illustration_and_writing}
The five years between 1943 and 1948 were some of the most productive of his career. He finished *Titus Groan* and *Gormenghast* and completed some of his most acclaimed illustrations for books by other authors, including Lewis Carroll\'s *The Hunting of the Snark* (for which he was reportedly paid only £5) and *Alice in Wonderland*, Samuel Taylor Coleridge\'s *The Rime of the Ancient Mariner*, the Brothers Grimm\'s *Household Tales*, *All This and Bevin Too* by Quentin Crisp and Robert Louis Stevenson\'s *Strange Case of Dr Jekyll and Mr Hyde*, as well as producing many original poems, drawings, and paintings.
Peake designed the logo for Pan Books. The publishers offered him either a flat fee of £10 or a royalty of one farthing per book. On the advice of Graham Greene, who told him that paperback books were a passing fad that would not last, Peake opted for the £10.
A book of nonsense poems, *Rhymes Without Reason*, was published in 1944 and was described by John Betjeman as \"outstanding\". Shortly after the war ended in 1945, Edgar Ainsworth, the art editor of *Picture Post*, commissioned Peake to visit France and Germany for the magazine. With writer Tom Pocock, Peake was among the first British civilians to witness the horrors of the Nazi concentration camp at Belsen, where the remaining prisoners, too sick to be moved, were dying before his very eyes. He made several drawings, but not surprisingly he found the experience profoundly harrowing, and expressed in deeply felt poems the ambiguity of turning their suffering into art.
In 1946, the family moved to Sark, where Peake continued to write and illustrate, and Maeve painted. *Gormenghast* was published in 1950, and the family moved back to England, settling in Smarden, Kent. Peake taught part-time at the Central School of Art, began his comic novel *Mr Pye*, and renewed his interest in theatre. His father died that year and left his house in Hillside Gardens in Wallington, Surrey to Peake. *Mr Pye* was published in 1953, and he later adapted it as a radio play. The BBC broadcast other plays of his in 1954 and 1956.
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# Mervyn Peake
## Later life {#later_life}
In 1956, Mervyn and Maeve visited Spain, financed by a friend who hoped that Peake\'s health, which was already declining, would be improved by the holiday. That year his novella *Boy in Darkness* was published beside stories by William Golding and John Wyndham in a volume called *Sometime, Never*. On 18 December the BBC broadcast his radio play *The Eye of the Beholder* (later revised as *The Voice of One*), in which an avant-garde artist is commissioned to paint a church mural. Peake placed much hope in his play *The Wit to Woo*, which was finally staged in London\'s West End in 1957, but it was a critical and commercial failure. This affected him greatly -- his health degenerated rapidly and he was again admitted to hospital with a nervous breakdown. During this period he was published primarily in New Worlds by Michael Moorcock, a consistent supporter since the mid-1950s.
### Declining health {#declining_health}
He was showing unmistakable early symptoms of dementia, for which he was given electroconvulsive therapy, to little avail. Over the next few years he gradually lost the ability to draw steadily and quickly, although he still managed to produce some drawings with the help of his wife. Among his last completed works were the illustrations for Balzac\'s *Droll Stories* (1961) and for his own poem *The Rhyme of the Flying Bomb* (1962), which he had written some 15 years earlier.
*Titus Alone* was published in 1959 and was revised in 1970 by Langdon Jones, an editor of *New Worlds*, to remove apparent inconsistencies introduced by the publisher\'s careless editing. Jones, also a composer, set *The Rhyme of the Flying Bomb* to music. A 1995 edition of all three completed Gormenghast novels includes a very short fragment of the beginning of what would have been the fourth Gormenghast novel, *Titus Awakes*, as well as a listing of events and themes he wanted to address in that and later Gormenghast novels.
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# Mervyn Peake
## Death
Throughout the 1960s, Peake\'s health declined into physical and mental incapacitation, and he died on 17 November 1968 at a care home run by his brother-in-law, at Burcot, near Oxford. He was buried in the churchyard of St Mary\'s in the village of Burpham, West Sussex.
A 2003 study published in *JAMA Neurology* assessed that Peake\'s death was the result of dementia with Lewy bodies (DLB).
His work, especially the *Gormenghast* series, became much better known and more widely appreciated after his death. They have since been translated into more than two dozen languages.
## Publications
Six volumes of Peake\'s verse were published during his lifetime; *Shapes & Sounds* (1941), *Rhymes without Reason* (1944), *The Glassblowers* (1950), *The Rhyme of the Flying Bomb* (1962), *Poems & Drawings* (1965), and *A Reverie of Bone* (1967). After his death came *Selected Poems* (1972), followed by *Peake\'s Progress* in 1979 -- though the Penguin edition of 1982, with many corrections, including a whole stanza inadvertently omitted from the hardback edition. *The Collected Poems of Mervyn Peake* was published by Carcanet Press in June 2008. Other collections include *The Drawings of Mervyn Peake* (1974), *Writings and Drawings* (1974), and *Mervyn Peake: the man and his art* (2006). A limited edition of the collected works, issued to celebrate Peake\'s centenary year, was published by Queen Anne Press.
## Archive
In 2010 an archive consisting of 28 containers of material, which included correspondence between Peake and Laurie Lee, Walter de la Mare and C. S. Lewis, plus 39 Gormenghast notebooks and original drawings for both *Alice Through the Looking Glass* and *Alice\'s Adventures in Wonderland*, was acquired by the British Library. Access to the Archive is available through the British Library website. In July 2020, the British Library acquired from the Peake Estate a visual archive consisting of 300 of Peake\'s original illustrations for children\'s stories, *Gormenghast*, and other works including *Treasure Island*.
## Commemoration
Peake\'s three children presented on BBC Radio Four in 2018 a half-hour memoir of their father\'s life, emphasizing the importance of the island of Sark.
The first blue plaque on Sark was unveiled in Peake\'s honour at the Gallery Stores in the Avenue on 30 August 2019.
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# Mervyn Peake
## Dramatic adaptations of Peake\'s work {#dramatic_adaptations_of_peakes_work}
In 1983, the Australian Broadcasting Corporation broadcast eight hour-long episodes for radio dramatising the complete Gormenghast Trilogy. This was the first to include the third book *Titus Alone*.
In 1984, BBC Radio 4 broadcast two 90-minute plays based on *Titus Groan* and *Gormenghast*, adapted by Brian Sibley and starring Sting as Steerpike and Freddie Jones as the Artist (narrator). A slightly abridged compilation of the two, running to 160 minutes, and entitled *Titus Groan of Gormenghast*, was broadcast on Christmas Day, 1992. BBC 7 repeated the original versions on 21 and 28 September 2003.
In 1986, *Mr Pye* was adapted as a four-part Channel 4 miniseries starring Derek Jacobi.
In 2000, the BBC and WGBH Boston co-produced a lavish miniseries, titled *Gormenghast*, based on the first two books of the series. It starred Jonathan Rhys-Meyers as Steerpike, Neve McIntosh as Fuchsia, June Brown as Nannie Slagg, Ian Richardson as Lord Groan, Christopher Lee as Flay, Richard Griffiths as Swelter, Warren Mitchell as Barquentine, Celia Imrie as Countess Gertrude, Lynsey Baxter and Zoë Wanamaker as the twins Cora and Clarice, and John Sessions as Dr Prunesquallor. The supporting cast included Olga Sosnovska, Stephen Fry and Eric Sykes, and the series is also notable as the last screen performance by comedy legend Spike Milligan (as the Headmaster).
A 30-minute TV short film entitled *A Boy in Darkness* (also made in 2000 and adapted from Peake\'s novella) was the first production from the BBC Drama Lab. It was set in a \"virtual\" computer-generated world created by young computer game designers, and starred Jack Ryder (from *EastEnders*) as Titus, with Terry Jones (*Monty Python\'s Flying Circus*) narrating.
Irmin Schmidt, founder of seminal German Krautrock group Can, wrote an opera called *Gormenghast*, based on the novels; it was first performed in Wuppertal, Germany, in November 1998. A number of early songs by New Zealand rock group Split Enz were inspired by Peake\'s work. The song \"The Drowning Man\", by British band The Cure, is inspired by events in *Gormenghast*, and the song \"Lady Fuchsia\" by another British band, Strawbs, is also based on events in the novels.
Peake\'s play *The Cave*, which dates from the mid-1950s, was given a first public reading at the Blue Elephant Theatre in Camberwell (London) in 2009, and had its world premiere in the same theatre, directed by Aaron Paterson, on 19 October 2010.
In 2011, Brian Sibley adapted the story again, this time as six one-hour episodes broadcast on BBC Radio 4 as the Classic Serial starting on 10 July 2011. The serial was titled *The History of Titus Groan* and adapted all three novels written by Mervyn Peake and the recently discovered concluding volume, *Titus Awakes*, completed by his widow, Maeve Gilmore. It starred Luke Treadaway as Titus, David Warner as the Artist and Carl Prekopp as Steerpike. It also starred Paul Rhys, Miranda Richardson, James Fleet, Tamsin Greig, Fenella Woolgar, Adrian Scarborough and Mark Benton among others.
Sting owned the film rights to the *Gormenghast* novels for a brief period in the 1980s, during which he discussed the possibility of adapting the novels into a series of concept albums, but he abandoned the idea after declaring the Radio 4 audio drama as ideal. As of 2015, author Neil Gaiman was in talks to adapt the novels for the big screen.
## Legacy
Authors who have cited Peake as influences on their work include: Neil Gaiman, Joanne Harris, Simon Maginn, Christopher Fowler and Susanna Clarke.
Peake is considered to be one of the Big Three of (secondary world) Fantasy, along with J. R. R. Tolkien and Robert E. Howard. Their equivalents in the science fiction genre are Isaac Asimov, Arthur C. Clarke, and Robert A. Heinlein
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# Finitary relation
In mathematics, a **finitary relation** over a sequence of sets `{{nowrap|''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>}}`{=mediawiki} is a subset of the Cartesian product `{{nowrap|''X''<sub>1</sub> × ... × ''X''<sub>''n''</sub>}}`{=mediawiki}; that is, it is a set of *n*-tuples `{{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}}`{=mediawiki}, each being a sequence of elements *x*~*i*~ in the corresponding *X*~*i*~. Typically, the relation describes a possible connection between the elements of an *n*-tuple. For example, the relation \"*x* is divisible by *y* and *z*\" consists of the set of 3-tuples such that when substituted to *x*, *y* and *z*, respectively, make the sentence true.
The non-negative integer *n* that gives the number of \"places\" in the relation is called the *arity*, *adicity* or *degree* of the relation. A relation with *n* \"places\" is variously called an ***n*-ary relation**, an ***n*-adic relation** or a **relation of degree *n***. Relations with a finite number of places are called *finitary relations* (or simply *relations* if the context is clear). It is also possible to generalize the concept to *infinitary relations* with infinite sequences.
## Definitions
Definition : *R* is an *n*-ary **relation** on sets `{{nowrap|''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>}}`{=mediawiki} is given by a subset of the Cartesian product `{{nowrap|''X''<sub>1</sub> × ... × ''X''<sub>''n''</sub>}}`{=mediawiki}.
Since the definition is predicated on the underlying sets `{{nowrap|''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>}}`{=mediawiki}, *R* may be more formally defined as the (`{{nowrap|''n'' + 1}}`{=mediawiki})-tuple `{{nowrap|(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>, ''G'')}}`{=mediawiki}, where *G*, called the *graph* of *R*, is a subset of the Cartesian product `{{nowrap|''X''<sub>1</sub> × ... × ''X''<sub>''n''</sub>}}`{=mediawiki}.
As is often done in mathematics, the same symbol is used to refer to the mathematical object and an underlying set, so the statement `{{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) ∈ ''R''}}`{=mediawiki} is often used to mean `{{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) ∈ ''G''}}`{=mediawiki} is read \"*x*~1~, \..., *x*~*n*~ are *R*-related\" and are denoted using prefix notation by `{{nowrap|''Rx''<sub>1</sub>⋯''x''<sub>''n''</sub>}}`{=mediawiki} and using postfix notation by `{{nowrap|''x''<sub>1</sub>⋯''x''<sub>''n''</sub>''R''}}`{=mediawiki}. In the case where *R* is a binary relation, those statements are also denoted using infix notation by `{{nowrap|''x''<sub>1</sub>''Rx''<sub>2</sub>}}`{=mediawiki}.
The following considerations apply:
- The set *X*~*i*~ is called the `{{itco|{{mvar|i}}}}`{=mediawiki}th *domain* of *R*. In the case where *R* is a binary relation, *X*~1~ is also called simply the *domain* or *set of departure* of *R*, and *X*~2~ is also called the *codomain* or *set of destination* of *R*.
- When the elements of *X*~*i*~ are relations, *X*~*i*~ is called a *nonsimple domain* of *R*.
- The set of `{{nowrap|∀''x''<sub>''i''</sub> ∈ ''X''<sub>''i''</sub>}}`{=mediawiki} such that `{{nowrap|''Rx''<sub>1</sub>⋯''x''<sub>''i''−1</sub>''x''<sub>''i''</sub>''x''<sub>''i''+1</sub>⋯''x''<sub>''n''</sub>}}`{=mediawiki} for at least one `{{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}}`{=mediawiki} is called the *i*th *domain of definition* or *active domain* of *R*. In the case where *R* is a binary relation, its first domain of definition is also called simply the *domain of definition* or *active domain* of *R*, and its second domain of definition is also called the *codomain of definition* or *active codomain* of *R*.
- When the `{{mvar|i}}`{=mediawiki}th domain of definition of *R* is equal to *X*~*i*~, *R* is said to be *total* on its *i*th domain (or on *X*~*i*~, when this is not ambiguous). In the case where *R* is a binary relation, when *R* is total on *X*~1~, it is also said to be *left-total* or *serial*, and when *R* is total on *X*~2~, it is also said to be *right-total* or *surjective*.
- When `{{nowrap|∀''x'' ∀''y'' ∈ ''X''<sub>''i''</sub>.}}`{=mediawiki} `{{nowrap|∀''z'' ∈ ''X''<sub>''j''</sub>.}}`{=mediawiki} `{{nowrap|1=''xR''<sub>''ij''</sub>''z'' ∧ ''yR''<sub>''ij''</sub>''z'' ⇒ ''x'' = ''y''}}`{=mediawiki}, where `{{nowrap|''i'' ∈ ''I''}}`{=mediawiki}, `{{nowrap|''j'' ∈ ''J''}}`{=mediawiki}, `{{nowrap|1=''R''<sub>''ij''</sub> = ''π''<sub>''ij''</sub> ''R''}}`{=mediawiki}, and `{{nowrap|{{mset|''I'', ''J''}}}}`{=mediawiki} is a partition of `{{nowrap|{{mset|1, ..., ''n''}}}}`{=mediawiki}, *R* is said to be *unique* on `{{nowrap|{{mset|''X''<sub>''i''</sub>}}<sub>''i''∈''I''</sub>}}`{=mediawiki}, and `{{nowrap|{{mset|''X''<sub>''i''</sub>}}<sub>''i''∈''J''</sub>}}`{=mediawiki} is called *a primary key* of *R*. In the case where *R* is a binary relation, when *R* is unique on `{{mset|''X''<sub>1</sub>}}`{=mediawiki}, it is also said to be *left-unique* or *injective*, and when *R* is unique on `{{mset|''X''<sub>2</sub>}}`{=mediawiki}, it is also said to be *univalent* or *right-unique*.
- When all *X*~*i*~ are the same set *X*, it is simpler to refer to *R* as an *n*-ary relation over *X*, called a *homogeneous relation*. Without this restriction, *R* is called a *heterogeneous relation*.
- When any of *X*~*i*~ is empty, the defining Cartesian product is empty, and the only relation over such a sequence of domains is the empty relation `{{nowrap|1=''R'' = ∅}}`{=mediawiki}.
Let a Boolean domain *B* be a two-element set, say, `{{nowrap|1=''B'' = {{mset|0, 1}}}}`{=mediawiki}, whose elements can be interpreted as logical values, typically `{{nowrap|1=0 = false}}`{=mediawiki} and `{{nowrap|1=1 = true}}`{=mediawiki}. The characteristic function of *R*, denoted by *χ*~*R*~, is the Boolean-valued function `{{nowrap|''χ''<sub>''R''</sub>: ''X''<sub>1</sub> × ... × ''X''<sub>''n''</sub> → ''B''}}`{=mediawiki}, defined by `{{nowrap|1=''χ''<sub>''R''</sub>({{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}}) = 1}}`{=mediawiki} if `{{nowrap|''Rx''<sub>1</sub>⋯''x''<sub>''n''</sub>}}`{=mediawiki} and `{{nowrap|1=''χ''<sub>''R''</sub>({{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}}) = 0}}`{=mediawiki} otherwise.
In applied mathematics, computer science and statistics, it is common to refer to a Boolean-valued function as an *n*-ary *predicate*. From the more abstract viewpoint of formal logic and model theory, the relation *R* constitutes a *logical model* or a *relational structure*, that serves as one of many possible interpretations of some *n*-ary predicate symbol.
Because relations arise in many scientific disciplines, as well as in many branches of mathematics and logic, there is considerable variation in terminology. Aside from the set-theoretic extension of a relational concept or term, the term \"relation\" can also be used to refer to the corresponding logical entity, either the logical comprehension, which is the totality of intensions or abstract properties shared by all elements in the relation, or else the symbols denoting these elements and intensions. Further, some writers of the latter persuasion introduce terms with more concrete connotations (such as \"relational structure\" for the set-theoretic extension of a given relational concept).
## Specific values of *n* {#specific_values_of_n}
### Nullary
Nullary (0-ary) relations count only two members: the empty nullary relation, which never holds, and the universal nullary relation, which always holds. This is because there is only one 0-tuple, the empty tuple (), and there are exactly two subsets of the (singleton) set of all 0-tuples. They are sometimes useful for constructing the base case of an induction argument.
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# Finitary relation
## Specific values of *n* {#specific_values_of_n}
### Unary
Unary (1-ary) relations can be viewed as a collection of members (such as the collection of Nobel laureates) having some property (such as that of having been awarded the Nobel Prize).
Every nullary function is a unary relation.
### Binary
Binary (2-ary) relations are the most commonly studied form of finitary relations. Homogeneous binary relations (where `{{nowrap|1=''X''<sub>1</sub> = ''X''<sub>2</sub>}}`{=mediawiki}) include
- Equality and inequality, denoted by signs such as = and \< in statements such as \"`{{nowrap|5 < 12}}`{=mediawiki}\", or
- Divisibility, denoted by the sign \| in statements such as \"`{{nowrap|13 {{!}}`{=mediawiki} 143}}\".
Heterogeneous binary relations include
- Set membership, denoted by the sign ∈ in statements such as \"`{{nowrap|1 ∈ '''N'''}}`{=mediawiki}\".
### Ternary
Ternary (3-ary) relations include, for example, the binary functions, which relate two inputs and the output. All three of the domains of a homogeneous ternary relation are the same set.
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# Finitary relation
## Example
Consider the ternary relation *R* \"*x* thinks that *y* likes *z*\" over the set of people `{{nowrap|1=''P'' = {{mset| Alice, Bob, Charles, Denise }}}}`{=mediawiki}, defined by:
: .
*R* can be represented equivalently by the following table:
*x* *y* *z*
--------- --------- --------
Alice Bob Denise
Charles Alice Bob
Charles Charles Alice
Denise Denise Denise
: Relation *R* \"*x* thinks that *y* likes *z*\"
Here, each row represents a triple of *R*, that is it makes a statement of the form \"*x* thinks that *y* likes *z*\". For instance, the first row states that \"Alice thinks that Bob likes Denise\". All rows are distinct. The ordering of rows is insignificant but the ordering of columns is significant.
The above table is also a simple example of a relational database, a field with theory rooted in relational algebra and applications in data management. Computer scientists, logicians, and mathematicians, however, tend to have different conceptions what a general relation is, and what it is consisted of. For example, databases are designed to deal with empirical data, which is by definition finite, whereas in mathematics, relations with infinite arity (i.e., infinitary relation) are also considered.
## History
The logician Augustus De Morgan, in work published around 1860, was the first to articulate the notion of relation in anything like its present sense. He also stated the first formal results in the theory of relations (on De Morgan and relations, see Merrill 1990).
Charles Peirce, Gottlob Frege, Georg Cantor, Richard Dedekind and others advanced the theory of relations. Many of their ideas, especially on relations called orders, were summarized in *The Principles of Mathematics* (1903) where Bertrand Russell made free use of these results.
In 1970, Edgar Codd proposed a relational model for databases, thus anticipating the development of data base management systems
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# Intuitionism
In the philosophy of mathematics, **intuitionism**, or **neointuitionism** (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
## Truth and proof {#truth_and_proof}
The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer\'s original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that construction by intuition. The vagueness of the intuitionistic notion of truth often leads to misinterpretations about its meaning. Kleene formally defined intuitionistic truth from a realist position, yet Brouwer would likely reject this formalization as meaningless, given his rejection of the realist/Platonist position. Intuitionistic truth therefore remains somewhat ill-defined. However, because the intuitionistic notion of truth is more restrictive than that of classical mathematics, the intuitionist must reject some assumptions of classical logic to ensure that everything they prove is in fact intuitionistically true. This gives rise to intuitionistic logic.
To an intuitionist, the claim that an object with certain properties exists is a claim that an object with those properties can be constructed. Any mathematical object is considered to be a product of a construction of a mind, and therefore, the existence of an object is equivalent to the possibility of its construction. This contrasts with the classical approach, which states that the existence of an entity can be proved by refuting its nonexistence. For the intuitionist, this is not valid; the refutation of the nonexistence does not mean that it is possible to find a construction for the putative object, as is required in order to assert its existence. As such, intuitionism is a variety of mathematical constructivism; but it is not the only kind.
The interpretation of negation is different in intuitionist logic than in classical logic. In classical logic, the negation of a statement asserts that the statement is *false*; to an intuitionist, it means the statement is *refutable*. There is thus an asymmetry between a positive and negative statement in intuitionism. If a statement *P* is provable, then *P* certainly cannot be refutable. But even if it can be shown that *P* cannot be refuted, this does not constitute a proof of *P*. Thus *P* is a stronger statement than *not-not-P*.
Similarly, to assert that *A* or *B* holds, to an intuitionist, is to claim that either *A* or *B* can be *proved*. In particular, the law of excluded middle, \"*A* or not *A*\", is not accepted as a valid principle. For example, if *A* is some mathematical statement that an intuitionist has not yet proved or disproved, then that intuitionist will not assert the truth of \"*A* or not *A*\". However, the intuitionist will accept that \"*A* and not *A*\" cannot be true. Thus the connectives \"and\" and \"or\" of intuitionistic logic do not satisfy de Morgan\'s laws as they do in classical logic.
Intuitionistic logic substitutes constructability for abstract truth and is associated with a transition from the proof of model theory to abstract truth in modern mathematics. The logical calculus preserves justification, rather than truth, across transformations yielding derived propositions. It has been taken as giving philosophical support to several schools of philosophy, most notably the Anti-realism of Michael Dummett. Thus, contrary to the first impression its name might convey, and as realized in specific approaches and disciplines (e.g. Fuzzy Sets and Systems), intuitionist mathematics is more rigorous than conventionally founded mathematics, where, ironically, the foundational elements which intuitionism attempts to construct/refute/refound are taken as intuitively given.
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# Intuitionism
## Infinity
Among the different formulations of intuitionism, there are several different positions on the meaning and reality of infinity.
The term potential infinity refers to a mathematical procedure in which there is an unending series of steps. After each step has been completed, there is always another step to be performed. For example, consider the process of counting: $1, 2, ...$
The term actual infinity refers to a completed mathematical object which contains an infinite number of elements. An example is the set of natural numbers, $\mathbb{N} = \{1, 2, ...\}$.
In Cantor\'s formulation of set theory, there are many different infinite sets, some of which are larger than others. For example, the set of all real numbers $\mathbb{R}$ is larger than $\mathbb{N}$, because any attempt to put the natural numbers into one-to-one correspondence with the real numbers will always fail: there will always be an infinite number of real numbers \"left over\". Any infinite set that can be placed in one-to-one correspondence with the natural numbers is said to be \"countable\" or \"denumerable\". Infinite sets larger than this are said to be \"uncountable\".
Cantor\'s set theory led to the axiomatic system of Zermelo--Fraenkel set theory (ZFC), now the most common foundation of modern mathematics. Intuitionism was created, in part, as a reaction to Cantor\'s set theory.
Modern constructive set theory includes the axiom of infinity from ZFC (or a revised version of this axiom) and the set $\mathbb{N}$ of natural numbers. Most modern constructive mathematicians accept the reality of countably infinite sets (however, see Alexander Esenin-Volpin for a counter-example).
Brouwer rejected the concept of actual infinity, but admitted the idea of potential infinity.
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# Intuitionism
## History
Intuitionism\'s history can be traced to two controversies in nineteenth century mathematics.
The first of these was the invention of transfinite arithmetic by Georg Cantor and its subsequent rejection by a number of prominent mathematicians including most famously his teacher Leopold Kronecker---a confirmed finitist.
The second of these was Gottlob Frege\'s effort to reduce all of mathematics to a logical formulation via set theory and its derailing by a youthful Bertrand Russell, the discoverer of Russell\'s paradox. Frege had planned a three-volume definitive work, but just as the second volume was going to press, Russell sent Frege a letter outlining his paradox, which demonstrated that one of Frege\'s rules of self-reference was self-contradictory. In an appendix to the second volume, Frege acknowledged that one of the axioms of his system did in fact lead to Russell\'s paradox.`{{refn| See {{harvnb|Frege|1960|pages=234–244}}}}`{=mediawiki}
Frege, the story goes, plunged into depression and did not publish the third volume of his work as he had planned. For more see Davis (2000) Chapters 3 and 4: Frege: *From Breakthrough to Despair* and Cantor: *Detour through Infinity.* See van Heijenoort for the original works and van Heijenoort\'s commentary.
These controversies are strongly linked as the logical methods used by Cantor in proving his results in transfinite arithmetic are essentially the same as those used by Russell in constructing his paradox. Hence how one chooses to resolve Russell\'s paradox has direct implications on the status accorded to Cantor\'s transfinite arithmetic.
In the early twentieth century L. E. J. Brouwer represented the *intuitionist* position and David Hilbert the formalist position---see van Heijenoort. Kurt Gödel offered opinions referred to as *Platonist* (see various sources re Gödel). Alan Turing considers: \"non-constructive systems of logic with which not all the steps in a proof are mechanical, some being intuitive\". Later, Stephen Cole Kleene brought forth a more rational consideration of intuitionism in his *Introduction to metamathematics* (1952).
Nicolas Gisin is adopting intuitionist mathematics to reinterpret quantum indeterminacy, information theory and the physics of time.
## Contributors
- Henri Poincaré (preintuitionism/conventionalism)
- L. E. J
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# May 6
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# Monkey Island
***Monkey Island*** is a series of adventure games. The first four games were produced and published by LucasArts, earlier known as Lucasfilm Games. The fifth was developed by Telltale Games with LucasArts, while the sixth was developed by Terrible Toybox with Lucasfilm Games and Devolver Digital.
The games follow the adventures of the hapless Guybrush Threepwood as he struggles to become the most notorious pirate in the Caribbean, defeat the plans of the evil undead pirate LeChuck and win the heart of Governor Elaine Marley. The plots often involve the mysterious Monkey Island and its secrets.
*Monkey Island* was created by Ron Gilbert. Gilbert worked on the first two games before leaving LucasArts. Dave Grossman and Tim Schafer, co-writers of the first two games, had success on other games before they both left LucasArts. The rights to *Monkey Island* remained with LucasArts, and the third and fourth games were created without direct involvement from the original writing staff. Grossman was a creative director on the fifth game in the series, which Gilbert was a consultant on the early stages of. Gilbert returned to the series with the sixth game, *Return to Monkey Island* (2022), which he co-wrote and co-designed with Grossman.
## Background
Ron Gilbert\'s two main inspirations for the story were Disneyland\'s Pirates of the Caribbean ride and Tim Powers\' book *On Stranger Tides*. The book was the inspiration for the story and characters, while the ride was the inspiration for the ambiance. Gilbert said in an interview that:
> \"\[the POTC Ride\] keeps you moving through the adventure but I\'ve always wished I could get off and wander around, learn more about the characters, and find a way onto those pirate ships. So with *The Secret of Monkey Island* I wanted to create a game that had the same flavor, but where you could step off the boat and enter that whole storybook world".
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# Monkey Island
## Media
### Games
#### *The Secret of Monkey Island* {#the_secret_of_monkey_island}
The series debuted in 1990 with *The Secret of Monkey Island* on the Amiga, MS-DOS, Atari ST and Macintosh platforms; the game was later ported to FM Towns and Mega-CD (1993). A remake version with updated graphics and new voiceovers was released for PlayStation Network, PC Windows, Xbox Live Arcade and OS X. An iPhone version was also released on July 23, 2009.
The game starts off with the main character Guybrush Threepwood stating \"I want to be a pirate!\" To do so, he must prove himself to three old pirate captains. During the perilous pirate trials, he meets the beautiful governor Elaine Marley, with whom he falls in love, unaware that the ghost pirate LeChuck also has his eyes on her. When Elaine is kidnapped, Guybrush procures crew and ship to track LeChuck down, defeat him and rescue his love.
#### *Monkey Island 2: LeChuck\'s Revenge* {#monkey_island_2_lechucks_revenge}
The second game, *Monkey Island 2: LeChuck\'s Revenge* from 1991, was available for fewer platforms; it was only released for PC MS-DOS, Amiga, Macintosh, and later for FM Towns. A Special Edition version, in a similar style as *The Secret of Monkey Island: Special Edition*, was released in July 2010 for iPhone, iPad, iPod Touch, Mac, PC, PS3 and Xbox 360.
As Guybrush, with a treasure chest in hand, and Elaine hang onto ropes in a void, he tells her the story of the game. He has decided to find the greatest of all treasures, that of Big Whoop. Unwittingly he helps revive LeChuck, who is now in zombie form. Guybrush is eventually captured by his nemesis, but escapes with help from Wally and finds the treasure only to find himself dangling from a rope, as depicted at the beginning of the game. As Guybrush concludes his story, his rope breaks and he finds himself facing LeChuck, whom he finally defeats using voodoo. The surrealistic ending is open to a number of interpretations. In the manual of *The Curse of Monkey Island*, it is stated that Guybrush falls victim to a hex implemented by LeChuck.
#### *The Curse of Monkey Island* {#the_curse_of_monkey_island}
*The Curse of Monkey Island*, the third in the series, was released exclusively for Microsoft Windows on PC in 1997, after a 6-year hiatus. *The Curse of Monkey Island* was released at the height of some of the biggest technological advancements in the gaming industry---digital audio, CD-ROM technology, and improved graphics.
*Monkey Island I* and *II* were originally released on floppy disks with text dialogue only. Entire conversations between characters would appear as written text, or as captions above their heads. The visuals of the third installment were also an improvement over the original game, using a more modern cel animation style. *The Curse of Monkey Island* is the only game in the series to feature this style of animation; subsequent games used three-dimensional polygon animation.
Threepwood, unwittingly, turns Elaine into a gold statue with a cursed ring, and she is subsequently stolen by pirates. He tracks her down before searching for another ring that can lift the curse. LeChuck appears in a fiery demon form, and is hot on Threepwood's heels until a stand-off on LeChuck\'s amusement park ride, Monkey Mountain.
#### *Escape from Monkey Island* {#escape_from_monkey_island}
*Escape from Monkey Island*, the fourth installment, was released in 2000 for PC Windows, and in 2001 for Macintosh and PlayStation 2.
When Guybrush Threepwood and Elaine Marley return from their honeymoon, they find that Elaine has been declared officially dead, her mansion is under a destruction order, and her position as governor is up for election. Guybrush investigates and unearths a conspiracy by LeChuck and evil real estate developer Ozzie Mandrill to use a voodoo talisman, \"The Ultimate Insult\", to make all pirates docile in order to turn the Caribbean into a center of tourism.
#### *Tales of Monkey Island* {#tales_of_monkey_island}
*Tales of Monkey Island* is the fifth installment within the series, co-developed by Telltale Games and LucasArts, with a simultaneous release both on WiiWare and PC. Unlike other installments, *Tales* is an episodic adventure consisting of five different episodes. The first episode was released on July 7, with the last one released on December 8, 2009.
During a heated battle with his nemesis, the evil pirate LeChuck, Guybrush unwittingly unleashes an insidious pox that rapidly spreads across the Caribbean, turning pirates into zombie-like monsters. The Voodoo Lady sends Guybrush in search of a legendary sea sponge to stem the epidemic, but this seemingly straightforward quest has surprises around every corner.
#### *Return to Monkey Island* {#return_to_monkey_island}
With the purchase of LucasArts by the Walt Disney Company in 2012, the rights to the franchise are now property of Disney. In the second half of 2010s, Disney Interactive ceased the production on gaming and transitioned to a licensing model. Gilbert wrote on Twitter that he was interested in buying the *Monkey Island* and *Maniac Mansion* properties. Fans of the series launched an online petition asking Disney to sell the franchise to Gilbert; by December 2021, the petition had gathered about 29,000 signatures.
*Return to Monkey Island*, the sixth *Monkey Island* installment, was released on September 19, 2022 on the Nintendo Switch and Windows, coming to other formats later. It is a collaboration between Gilbert\'s Terrible Toybox studio and Lucasfilm Games, and published by Devolver Digital. A frame story in the game serves to explain and continue from the ending of *LeChuck\'s Revenge*, while the main narrative takes place after the other games in the series. Ron Gilbert has expressed his desire to tell a simple and focused pirate story in the game, while also redefining the adventure game user interface and deepening the greater lore. In addition to Gilbert, Grossman returns as co-writer, with music from veteran series composers Michael Land, Peter McConnell, and Clint Bajakian, and Dominic Armato, Alexandra Boyd, and Denny Delk reprising their roles as Guybrush, Elaine, and Murray. Jess Harnell replaces the retired Earl Boen as the voice of LeChuck.
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# Monkey Island
## Media
### Games
#### Other appearances {#other_appearances}
Stan\'s Used Coffins is referred to in one of the levels of the LucasArts game *Outlaws*. In *Indiana Jones and the Infernal Machine*, Guybrush can be accessed as a playable character via a cheat code; in addition, a *Monkey Island*-themed secret room can be found in the game\'s final level. Guybrush also appears in *Star Wars: The Force Unleashed II* as a playable skin for Starkiller named \"Guybrush Threepkiller\".
Guybrush is paid homage in the Naughty Dog video game *Uncharted 4: A Thief\'s End*, where a pirate with major similarities to Guybrush is featured as one of the twelve pirate captains that founded Libertalia. Although he remains unnamed throughout the game, the resemblance is uncanny and his sigil is represented by a monkey. His portrait can be seen in the Libertalia treasury with the other founders and though his name is partly scratched out, the letters still visible spell out the truncated name \"Guy Wood\".
Several elements from the *Monkey Island* series appear in *Sea of Thieves* as part of its June 2021 \"A Pirate\'s Life\" update. Developed in collaboration with Disney and primarily themed after *Pirates of the Caribbean*, multiple references to the characters and locales from the Monkey Island franchise can be found in journals by Kate Capsize scattered around the wreckage of The Headless Monkey during the update\'s first Tall Tale, accompanied by an original arrangement of the Monkey Island theme. According to the journals, Guybrush and Elaine Threepwood are celebrating their honeymoon somewhere upon the Sea of Thieves, while Kate perished attempting to get revenge on Guybrush for framing her.
A full *Monkey Island*-themed expansion for the game, \"The Legend of Monkey Island\", was released on July 20, 2023 and spread across three monthly episodes. In the story, set between *Curse* and *Escape*, Guybrush and Elaine\'s honeymoon on the Sea of Thieves is interrupted by LeChuck, who traps them in a dream version of Mêlée Island where everyone worships Guybrush as a legendary pirate. To stop LeChuck from restoring the legendary Burning Blade and conquering the Sea of Thieves, the Pirate Lord recruits the now-revived Kate Capsize and the player pirates to enter the dreamworld and rescue Guybrush and Elaine.
In an update to *Hitman 3*, a new pirate-themed map was added, which featured an Easter Egg referencing *Monkey Island* in the form of a gravestone in the environment reading \"G Threepwood, Mighty Pirate\", a clear reference to Guybrush.
In *The Witcher 3: Wild Hunt*, while completing the \'Fists of Fury\' quest in the *Blood and Wine* expansion patch, the protagonist Geralt encounters a man named Mancomb, a reference to a character of the same name in the first game of the series.
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# Monkey Island
## Media
### Cancelled film {#cancelled_film}
Shortly after Pixar, a spinout from Lucasfilm, found success with the first *Toy Story* film in 1995, there had been a push across Hollywood for more digitally animated films. Lucasfilm\'s Industrial Light & Magic (ILM), in the midst of transitioning from practical to digital effects, offered its services for producing these films to other studios. One of the first projects they tried to work on was with Universal Pictures to revive the Universal Classic Monsters line with a film called *Frankenstein and the Wolfman*. While several scripts and preliminary art was produced for this film, shake-ups at Universal due to the financial failure of *Babe: Pig in the City* led to changes in leadership for the film and ultimately its cancellation.
David Carson, who had been set to direct *Frankenstein and the Wolfman* but left after the Universal shake-up, came back to ILM with the idea of an animated film based on the first *Monkey Island* game around 2000. With initial support from ILM, Carson worked an initial script with Corey Rosen and Scott Leberecht as to pitch the idea to Amblin Entertainment, the production company owned by Steven Spielberg. Spielberg had told Carson that he had previously told George Lucas that he should have made a *Monkey Island* movie years before, and other meetings with Amblin went well to proceed to further screenwriting work. The rest of ILM\'s story department was brought in to help write, including Steve Purcell, but this team worked separately from the writers that were developing the actual games, creating a disconnect between story the film was going with and the narrative already established in the video game series. As they continued to work out the screenplay, the direction of the film continued to veer further from the video game series, including at one point where Spielberg had suggested the game be about the monkeys on Monkey Island instead of the pirates. According to Carson, the lack of a creative direction at this point led to the film being shelved at ILM.
Details about the film were first revealed publicly in 2011 as part of the *Monkey Island Special Edition Collection* which included some of the film\'s concept art, storyboards, and scripts.
It had been rumored that Ted Elliott and Terry Rossio had been involved in the writing of the *Monkey Island* script which they subsequently used as the basis for the first *Pirates of the Caribbean* film. Both Elliott and Rossio had been to ILM and were shown parts of the *Monkey Island* script, around the same time they were working on their script for *Pirates*. When *Pirates* was released, many fans of the *Monkey Island* series made comparisons of parts of the film to the games, and when news of the cancelled film first arose in 2011, the potential connection of Elliott and Rossio to the *Monkey Island* script started. Both Carson and Rossio stated that many of the tropes in both *Monkey Island* and *Pirates* are based on the classic pirate movies and that there was no direct reuse of the cancelled *Monkey Island* film in *Pirates*.
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# Monkey Island
## The \"Secret\" of Monkey Island {#the_secret_of_monkey_island_1}
None of the first five games explicitly reveal the \"Secret of Monkey Island\". The team behind *Escape from Monkey Island* attempted to resolve the issue by showing that the Giant Monkey Head was actually the control room of a Giant Monkey Robot. The cut-scene in which the revelation was made is called \"The Real Secret of Monkey Island\".
Gilbert stated that he never told anyone what the true secret of Monkey Island is. In a 2004 interview, Gilbert stated that when the game was originally conceived, it was considered \"too big\", so they split it into three parts. He added that he \"knows what the third \[part\] is\" and \"how the story\'s supposed to end\", indicating that he had a definite concept of the "secret" and a conclusive third game.
The true nature of the secret is the main focus of *Return to Monkey Island*, with several characters competing amongst themselves in a race to discover \"the Secret\". The game\'s conclusion reveals the secret to be a novelty T-shirt earned as a prize at a pirate-themed amusement park, which has acted as the setting for all of Guybrush Threepwood's previous adventures. Threepwood, as the game\'s narrator, is intentionally ambiguous as to whether this is the actual secret, even suggesting that the secret means different things to different people, and putting forth the notion that the story of the journey (and the joy of speculating about the secret with others) is more valuable than the reward itself.
After the release of *Return to Monkey Island*, Gilbert stated in an interview that the true secret (as conceived during development of the first installment of the series) is that Guybrush was, in fact, inside of a pirate-themed amusement park the entire time
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# Cardiff Arms Park
**Cardiff Arms Park** (*Parc yr Arfau Caerdydd*), also known as **The Arms Park**, is primarily a rugby union stadium, and also has a bowling green. It is situated in Cardiff, Wales, next to the Millennium Stadium. The Arms Park was host to the British Empire and Commonwealth Games in 1958, and hosted four games in the 1991 Rugby World Cup, including the third-place play-off. The Arms Park also hosted the inaugural Heineken Cup Final of 1995--96 and the following year in 1996--97.
The history of the rugby ground begins with the first stands appearing for spectators in the ground in 1881--1882. Originally the Arms Park had a cricket ground to the north and a rugby union stadium to the south. By 1969, the cricket ground had been demolished to make way for the present day rugby ground to the north and a second rugby stadium to the south, called the National Stadium. The National Stadium, which was used by Wales national rugby union team, was officially opened on 7 April 1984, however in 1997 it was demolished to make way for the Millennium Stadium in 1999, which hosted the 1999 Rugby World Cup and became the national stadium of Wales. The rugby ground has remained the home of the semi-professional Cardiff RFC yet the professional Cardiff Blues regional rugby union team moved to the Cardiff City Stadium in 2009, but returned three years later.
The site is owned by Cardiff Athletic Club and has been host to many sports, apart from rugby union and cricket; they include athletics, association football, greyhound racing, tennis, British baseball and boxing. The site also has a bowling green to the north of the rugby ground, which is used by Cardiff Athletic Bowls Club, which is the bowls section of the Cardiff Athletic Club. The National Stadium also hosted many music concerts including Michael Jackson, Dire Straits, David Bowie, Bon Jovi, The Rolling Stones and U2.
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# Cardiff Arms Park
## History
### Early history of the site {#early_history_of_the_site}
The Cardiff Arms Park site was originally called the Great Park, a swampy meadow behind the Cardiff Arms Hotel. The hotel was built by Sir Thomas Morgan, during the reign of Charles I. Cardiff Arms Park was named after this hotel. From 1803, the Cardiff Arms Hotel and the Park had become the property of the Bute family. The Arms Park soon became a popular place for sporting events, and by 1848, Cardiff Cricket Club was using the site for its cricket matches. However, by 1878, Cardiff Arms Hotel had been demolished.
The 3rd Marquess of Bute stipulated that the ground could only be used for \"recreational purposes\". At that time Cardiff Arms Park had a cricket ground to the north and a rugby union ground to the south. 1881--2 saw the first stands for spectators; they held 300 spectators and cost £50. The architect was Archibald Leitch, who also designed Ibrox Stadium and Old Trafford. In 1890, new standing areas were constructed along the entire length of the ground, with additional stands erected in 1896.
### 1912 redevelopment
By 1912, the Cardiff Football Ground, as it was then known, had a new south stand and temporary stands on the north, east and west ends of the ground. The south stand was covered, while the north terrace was initially without a roof. The improvements were partly funded by the Welsh Rugby Union (WRU). The opening ceremony took place on 5 October 1912, with a match between Newport RFC and Cardiff RFC. The new ground was opened by Lord Ninian Crichton-Stuart. This new development increased the ground capacity to 43,000 and much improved facilities at the ground compared to the earlier stands.
In 1922, The 4th Marquess of Bute sold the entire site and it was bought by the Cardiff Arms Park Company Limited for £30,000. It was then leased to the Cardiff Athletic Club (cricket and rugby sections) for 99 years at a cost of £200 per annum.
### North and South Stand redevelopments {#north_and_south_stand_redevelopments}
During 1934 the cricket pavilion had been demolished to make way for the new North Stand which was built on the rugby union ground, costing around £20,000. However, in 1941 the new North Stand and part of the west terracing was badly damaged in the Blitz by the Luftwaffe during the Second World War.
At a general meeting of the WRU in June 1953 they made a decision \"That until such time as the facilities at Swansea were improved, all international matches be played at Cardiff\". At the same time, plans were made for a new South Stand which was estimated to cost £60,000; the tender price, however, came out at £90,000, so a compromise was made and it was decided to build a new upper South Stand costing £64,000 instead, with the Cardiff Athletic Club contributing £15,000 and the remainder coming from the WRU. The new South Stand opened in 1956, in time for the 1958 British Empire and Commonwealth Games. This brought the overall capacity of the Arms Park up to 60,000 spectators, of which 12,800 were seated and the remainder standing.
The Arms Park hosted the 1958 British Empire and Commonwealth Games, which was used for the athletics events, but this event caused damage to the drainage system, so much so, that other rugby unions (England, Scotland and Ireland) complained after the Games about the state of the pitch. On 4 December 1960, due to torrential rain, the River Taff burst its banks with the Arms Park pitch being left under 4 ft of water. The Development Committee was set up to resolve these issues on a permanent basis. They looked at various sites in Cardiff, but they all proved to be unsatisfactory. They also could not agree a solution with the Cardiff Athletic Club, so they purchased about 80 acre of land at Island Farm in Bridgend, which was previously used as a prisoner-of-war camp. It is best known for being the camp where the biggest escape attempt was made by German prisoners of war in Great Britain during the Second World War. Due to problems including transport issues Glamorgan County Council never gave outline planning permission for the proposals and by June 1964 the scheme was abandoned. At that stage, the cricket ground to the north was still being used by Glamorgan County Cricket Club, and the rugby union ground to the south was used by the national Wales team and Cardiff RFC.
By 7 October 1966, the first floodlit game was held at Cardiff Arms Park, a game in which Cardiff RFC beat the Barbarians by 12 points to 8.
### National Stadium redevelopment {#national_stadium_redevelopment}
The National Stadium, which was previously known as the Welsh National Rugby Ground, was designed by Osborne V Webb & Partners and built by G A Williamson & Associates of Porthcawl and Andrew Scott & Company of Port Talbot. In 1969 construction began on the stadium which replaced the existing rugby ground built in 1881. The stadium was home to the Wales national rugby union team since 1964 and the Wales national football team since 1989. In 1997 the stadium was demolished to make way for the new Millennium Stadium.
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# Cardiff Arms Park
## History
### Millennium Stadium {#millennium_stadium}
*Main article: Millennium Stadium*
Thirteen years after the National Stadium had opened in 1984, it was considered too small and did not have the facilities required of the time and it was demolished and a new stadium, the Millennium Stadium, was built in its place (completed to a north--south alignment and opened in June 1999). This would become the fourth redevelopment on the site.
Construction involved the demolition of a number of buildings, primarily the existing National Stadium, Wales Empire Pool in Wood Street, Cardiff Empire Telephone Exchange building and the newly built Territorial Auxiliary and Volunteer Reserve building both in Park Street, and the Social Security offices in Westgate Street. The Millennium Stadium is now on roughly two-thirds of the National Stadium, but it no longer uses the Arms Park name. Since 2016, it has been known as the Principality Stadium.
## Timeline
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# Cardiff Arms Park
## Current site {#current_site}
### Rugby ground {#rugby_ground}
Only the rugby ground and the Cardiff Athletic Bowls Club now use the name Cardiff Arms Park. The rugby ground has two main stands, the North Stand and the South Stand. Both the Stands have terracing below seating. The other stands in the ground are the Westgate Street end Family Stand, which has rows of seating below executive boxes, plus the club shop, and the River Taff end (the Barry Nelmes Suite, named after Barry Nelmes, the former Cardiff RFC captain), which has 26 executive boxes. The rugby ground has two main entrances, the south entrance, and the Gwyn Nicholls Memorial Gates (Angel Hotel entrance), which was unveiled on 26 December 1949 in honour of the Welsh international rugby player Gwyn Nicholls. The Cardiff Athletic Clubhouse is situated in the corner of the ground between the South Stand and the Westgate Street end.
The South Stand of the rugby ground formed a complete unit with the North Stand of the National Stadium. Now the same structure of the South Stand of the rugby ground is also physically attached to the North Stand of the Millennium Stadium. This section is known colloquially as Glanmor\'s Gap, after Glanmor Griffiths, former chair and President of the WRU. This came about because the WRU were unable to secure enough funding to include the North Stand in the Millennium Stadium, and the National Lottery Commission would not provide any additional funds to be used for the construction of a new ground for Cardiff RFC. The Millennium Stadium was therefore built with the old reinforced concrete structure of the National Stadium (North Stand) and the new steel Millennium Stadium structure built around it.
There was doubt about the future of the Arms Park after 2010 following the move of the Cardiff Blues to the Cardiff City Stadium. Cardiff RFC Ltd, the company that runs Cardiff Blues and Cardiff RFC, still has a 15-year lease on the Arms Park, but talks are underway to release the rugby club from the terms of the lease, to enable the Millennium Stadium to be redeveloped with a new North Stand and adjoining convention centre. However, it still has the original requirement on the lease, that the land will only be used for \"recreational purposes\", as stipulated by the Bute family. But the Arms Park site is a prime piece of real estate in the centre of Cardiff, which means that it may be difficult to sell the land to property developers. The estimated value of the whole Arms Park site could be at least £25 million, although with the \"recreational use\" requirement, its actual value could be a lot less than that figure. A decision by Cardiff Athletic Club on the future of the Arms Park has yet to be made. In 2011, the Cardiff Blues regional rugby union team made a £6 million bid for the Arms Park, later the WRU made an increased bid of £10 million for the site. Both bids were rejected by the trustees of the Cardiff Athletic Club. However, in 2012 Cardiff Blues announced that they would be making a permanent return to Cardiff Arms Park following declining attendances at the Cardiff City Stadium. During the 2013 off-season, the pitch at the rugby ground was replaced with an all weather 3G (third generation) artificial turf surface from FieldTurf at a cost of £400,000, intended to prevent any adverse weather conditions from affecting the rugby.
**Proposed redevelopment**
An agreement in principle was reached in December 2015 between the landlord of the stadium site (Cardiff Athletic Club) and its tenant (Cardiff Blues) to give the club a 150-year lease on the stadium site. This could see the redevelopment of the Arms Park, including a new 15,000 seater stadium at 90 degrees to the existing stadium costing between £20 million and £30 million and surrounded by new offices and apartments. If the final agreement goes ahead, Cardiff Athletic Club would receive an upfront payment of approximately £8 million. As part of the agreement, the bowls section would have to vacate its current site at the Arms Park and move to a new facility. At present Cardiff Blues pay Cardiff Athletic Club rent of around £115,000 per annum, however this would nearly double to around £200,000.
### Bowling green {#bowling_green}
Cardiff Arms Park is best known as a rugby union stadium, but Cardiff Athletic Bowls Club (CABC) was established in 1923, and ever since then, the club has used the Arms Park as its bowling green. The bowls club is a section of the Cardiff Athletic Club and shares many of the facilities of the Cardiff Arms Park athletics centre.
The Les Spence Memorial Gates were erected in memory of the former Cardiff RFU player, who captained the team in 1936--37. He was born in 1907 and became chairman of the Cardiff RFU and president of the WRU between 1973 and 1974. He was awarded an MBE and died in 1988.
The club has produced two Welsh international bowlers; Mr. C Standfast in 1937 and Mr. B Hawkins who represented Wales in the 1982 World Pairs and captained Wales in 1982 and 1984.
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# Cardiff Arms Park
## Usage
### Association football {#association_football}
The Riverside Football Club, founded in 1899, played some matches at the Arms Park until 1910, when they moved to Ninian Park, and later became Cardiff City Football Club.
On 31 May 1989, Wales played its first international game against West Germany at the National Stadium in a World Cup qualifying match, which ended goalless. It was also the first ever international football match held in Great Britain that was watched by all-seater spectators.
The adjoining Cardiff Rugby Club ground has also been used for Association Football. In July 1995, Ton Pentre played two Intertoto Cup games there, against Heerenveen (Netherlands) and Uniao Leiria (Portugal) as their own ground was not suitable. The Heerenveen game - the first ever soccer match to be played there - kicked off at 6pm on Saturday 1 July 1995 and resulted in the Dutch side winning 7--0. The Wales U-21 team have also played a home game there in the late 1990s.
On 5 April 2017, the ground was used to host the men\'s and women\'s football matches as part of the 2017 Welsh Varsity, between Cardiff University and Swansea University. The women\'s game finished in a 1-1 draw, while the men\'s game resulted in a 1-0 win for Swansea.
### Athletics
In 1958, the British Empire and Commonwealth Games were held in Cardiff. The event was (to date) the biggest sporting event ever held in Wales; however, it would not have been possible without the financial support given by the WRU and the Cardiff Athletic Club. Both the opening and closing ceremonies took place at Cardiff Arms Park, plus all the track and field events, on what had been the greyhound track. It would turn out to be the last time that South Africa would participate in the Games until 1994. South Africa withdrew from the Commonwealth Games in 1961.
### Baseball and British baseball {#baseball_and_british_baseball}
Baseball was established early on in Cardiff, and one of the earliest of games to be held at the Arms Park was on 18 May 1918. It was a charity match in aid of the Prisoner of War Fund between Welsh and American teams of the U.S. Beaufort and U.S. Jupiter. British baseball matches have also regularly taken place at the Arms Park and hosted the annual England versus Wales international game every four years. The games are now usually held at Roath Park.
### Boxing
The first boxing contest held at the Arms Park was on 24 January 1914, when Bombardier Billy Wells beat Gaston Pigot by a knockout in the first round of a 20-round contest. Boxing contests were held later on 14 June 1943, 12 August 1944, 4 October 1951 and 10 September 1952.
Around 25,000 spectators watched international boxing on 1 October 1993, at the National Stadium with a World Boxing Council (WBC) Heavyweight title bout between Lennox Lewis and Frank Bruno. It was the first time that two British-born boxers had fought for the world heavyweight title. Lewis beat Bruno by a technical knockout in the 7th round, in what was called the \"Battle of Britain\". On 30 September 1995, Steve Robinson the World Boxing Organization (WBO) World Featherweight Champion, lost against Prince Naseem Hamed at the rugby ground in 8 rounds.
### Cricket
In 1819, Cardiff Cricket Club was formed and by 1848 they had moved to their new home at the Arms Park. Glamorgan County Cricket Club, at the time not a first-class county, played their first match at the ground in June 1869 against Monmouthshire Cricket Club. The county club played their first County Championship match on the ground in 1921, competing there every season (except while first-class cricket was suspended during the Second World War) until their final match on the ground against Somerset in August 1966.
Cardiff Cricket Club played their final game at the ground against Lydney Cricket Club on 17 September 1966. Both Cardiff Cricket Club and Glamorgan then moved to a new ground at Sophia Gardens on the opposite bank of the River Taff to the Arms Park following work on the creation of the national rugby stadium.
The first first-class cricket match to be held on the ground was between West of England and East of England, on 20 June 1910. In all more than 240 first-class matches were played on the ground, all but two involving Glamorgan as the home team. Only one List A cricket match was played on the ground, Glamorgan\'s Gillette Cup fixture against Somerset on 22 May 1963.
### Greyhound racing {#greyhound_racing}
*Main article: Cardiff Greyhounds* Greyhound racing took place at the Arms Park for fifty years from 1927 until 1977.
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# Cardiff Arms Park
## Usage
### Rugby union {#rugby_union}
In 1876, the Cardiff RFC was formed and soon after they also used the park. On 12 April 1884, the first international match was played at the ground between Wales and Ireland, when 5,000 people watched Wales beat Ireland by two tries and a drop goal to nil.
The Arms Park rugby ground became the permanent home of the Wales national rugby union team in 1964. Later, the National Stadium was also home to the WRU Challenge Cup from 1972 until the match held at the Stadium on 26 April 1997, at a much reduced capacity, between Cardiff RFC and Swansea RFC. Cardiff RFC won the match 33--26.
The National Stadium is best known as the venue for what is considered to be \"the greatest try ever scored\" by Gareth Edwards for the Barbarians against New Zealand in what is also called \"the greatest match ever played\" on 27 January 1973. The final result was a win for the Barbarians. The score, 23--11, which translates to 27--13 in today\'s scoring system.
The scorers were:\
Barbarians: Tries: Gareth Edwards, Fergus Slattery, John Bevan, J P R Williams; Conversions: Phil Bennett (2); Penalty: Phil Bennett.\
All Blacks: Tries: Grant Batty (2); Penalty: Joseph Karam.
The National Stadium hosted four games in the 1991 Rugby World Cup, including the third-place play-off. The National Stadium was also host to the inaugural Heineken Cup final of 1995--96 when Toulouse beat Cardiff RFC by 21--18 after extra time, in front of 21,800 spectators. The following final in 1996--97 was also held at the National Stadium, this time it was between Brive and Leicester Tigers. Brive won the match 28--9, in front of a crowd of 41,664.
In 2008, the rugby ground hosted all the games in Pool A of the 2008 IRB Junior World Championship and also the semi-final on 18 June 2008, in which England beat South Africa 26--18.
Until February 2012, it had been assumed that the last professional rugby union game to take place at the Arms Park was on 17 May 2009, when Edinburgh beat the Cardiff Blues 36--14 in a Celtic League match during the 2008--09 season.
However, on Tuesday, 7 February 2012, it was confirmed that Cardiff Blues would face Connacht at the Arms Park on Friday, 10 February 2012. The Pro12 League game result was a win for the Cardiff Blues 22--15 and attendance of 8,000. The following Tuesday, it was announced that the match against Ulster on Friday, 17 February, would also be at the Arms Park, resulting in a Blues win, 21--14 and attendance of 8,600. The agreement signed during 2009 tied Cardiff Blues to a 20-year contract to play a maximum of 18 games per season for a set fee, rather than per match at Cardiff City Stadium. But on 23 February, it was announced that the two Welsh \'derbies\' against the Scarlets and the Ospreys would be played at Cardiff City Stadium, rather than the Arms Park, because of Cardiff Blues\' anticipation that the attendance figures would far exceed the maximum capacity of 9,000. On 8 May 2012, it was announced that Cardiff Blues would be returning to the Arms Park on a permanent basis after just three years at the Cardiff City Stadium.
On 23 May 2014, the rugby ground hosted the final of the 2013--14 Amlin Challenge Cup in which Northampton Saints beat Bath 30--16.
**Rugby World Cup**
Cardiff Arms Park hosted matches of the 1991 Rugby World Cup.
Date Competition Home team Away team
----------------- ------------------------------------------- ----------- ---- -----------
6 October 1991 1991 Rugby World Cup Pool 3 13
9 October 1991 1991 Rugby World Cup Pool 3 16
12 October 1991 1991 Rugby World Cup Pool 3 3
30 October 1991 1991 Rugby World Cup Third-place play-off 13
### Rugby league {#rugby_league}
South Wales Scorpions played a Rugby League Championship 1 match against London Skolars at Cardiff Arms Park on Sunday, 27 July 2014 and on Sunday 10 May 2015 at Cardiff Arms Park, South Wales Scorpions took on North Wales Crusaders. The 2015 European Cup match between France and Wales was held at Cardiff Arms Park on Friday on 30 October 2015.
On 11 April it was announced Cardiff Arms Park would be the new home ground of the Women\'s Betfred Super League South team Cardiff Demons. The inaugural league champions will play all home games at the stadium during the 2022 season.
The highest attendance for a rugby league game at the Arms Park was recorded on 8 June 1996 during the first Super League season when 6,708 saw St. Helens defeat the Sheffield Eagles 43--32. The St Helens team at the time contained Welsh players Anthony Sullivan, Karle Hammond and Keiron Cunningham.
**Rugby league test matches**
List of rugby league test matches played at Cardiff Arms Park.
Test# Date Result Attendance Notes
------- ----------------- --------------------------------- ------------ -----------------------------------------
1 26 June 1996 26--12 `{{rl|WAL}}`{=mediawiki} 5,425 1996 European Rugby League Championship
2 30 October 2015 14--6 `{{rl|FRA}}`{=mediawiki} 1,028 2015 European Cup
### Tennis
Tennis courts were laid out in the Arms Park for Cardiff Tennis Club until the club moved to Sophia Gardens in 1967. In 2003, the club amalgamated with Lisvane Tennis Club to form Lisvane (CAC) Tennis Club, which is still a section of Cardiff Athletic Club (CAC).
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# Cardiff Arms Park
## Usage
### Music concerts {#music_concerts}
Major music concerts were also held at the National Stadium from 1987 until 1996, they included Tina Turner, U2, Michael Jackson, The Rolling Stones, Dire Straits, Bon Jovi and R.E.M. The last music concert was held on 14 July 1996. Jehovah\'s Witnesses held their annual conventions at the National Stadium.
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# Cardiff Arms Park
## Singing tradition {#singing_tradition}
The National Stadium was known primarily as the venue for massed voices singing such hymns as \"Cwm Rhondda\", \"Calon Lân\", \"Men of Harlech\" and \"Hen Wlad Fy Nhadau\" (\"Land of my Fathers\" -- the national anthem of Wales). The legendary atmosphere including singing of the crowd was said to be worth at least a try or a goal to the home nation. This tradition of singing has now passed on to the Millennium Stadium.
The Arms Park has its own choir, called the Cardiff Arms Park Male Choir. It was formed in 1966 as the Cardiff Athletic Club Male Voice Choir, and today performs internationally with a schedule of concerts and tours. In 2000, the choir changed their name to become the Cardiff Arms Park Male Choir
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# Microevolution
**Microevolution** is the change in allele frequencies that occurs over time within a population. This change is due to four different processes: mutation, selection (natural and artificial), gene flow and genetic drift. This change happens over a relatively short (in evolutionary terms) amount of time compared to the changes termed macroevolution.
Population genetics is the branch of biology that provides the mathematical structure for the study of the process of microevolution. Ecological genetics concerns itself with observing microevolution in the wild. Typically, observable instances of evolution are examples of microevolution; for example, bacterial strains that have antibiotic resistance.
Microevolution provides the raw material for macroevolution.
## Difference from macroevolution {#difference_from_macroevolution}
Macroevolution is guided by sorting of interspecific variation (\"species selection\"), as opposed to sorting of intraspecific variation in microevolution. Species selection may occur as (a) effect-macroevolution, where organism-level traits (aggregate traits) affect speciation and extinction rates, and (b) strict-sense species selection, where species-level traits (e.g. geographical range) affect speciation and extinction rates. Macroevolution does not produce evolutionary novelties, but it determines their proliferation within the clades in which they evolved, and it adds species-level traits as non-organismic factors of sorting to this process.
## Four processes {#four_processes}
### Mutation
Mutations are changes in the DNA sequence of a cell\'s genome and are caused by radiation, viruses, transposons and mutagenic chemicals, as well as errors that occur during meiosis or DNA replication. Errors are introduced particularly often in the process of DNA replication, in the polymerization of the second strand. These errors can also be induced by the organism itself, by cellular processes such as hypermutation. Mutations can affect the phenotype of an organism, especially if they occur within the protein coding sequence of a gene. Error rates are usually very low---1 error in every 10--100 million bases---due to the proofreading ability of DNA polymerases. (Without proofreading error rates are a thousandfold higher; because many viruses rely on DNA and RNA polymerases that lack proofreading ability, they experience higher mutation rates.) Processes that increase the rate of changes in DNA are called mutagenic: mutagenic chemicals promote errors in DNA replication, often by interfering with the structure of base-pairing, while UV radiation induces mutations by causing damage to the DNA structure. Chemical damage to DNA occurs naturally as well, and cells use DNA repair mechanisms to repair mismatches and breaks in DNA---nevertheless, the repair sometimes fails to return the DNA to its original sequence.
In organisms that use chromosomal crossover to exchange DNA and recombine genes, errors in alignment during meiosis can also cause mutations. Errors in crossover are especially likely when similar sequences cause partner chromosomes to adopt a mistaken alignment making some regions in genomes more prone to mutating in this way. These errors create large structural changes in DNA sequence---duplications, inversions or deletions of entire regions, or the accidental exchanging of whole parts between different chromosomes (called translocation).
Mutation can result in several different types of change in DNA sequences; these can either have no effect, alter the product of a gene, or prevent the gene from functioning. Studies in the fly *Drosophila melanogaster* suggest that if a mutation changes a protein produced by a gene, this will probably be harmful, with about 70 percent of these mutations having damaging effects, and the remainder being either neutral or weakly beneficial. Due to the damaging effects that mutations can have on cells, organisms have evolved mechanisms such as DNA repair to remove mutations. Therefore, the optimal mutation rate for a species is a trade-off between costs of a high mutation rate, such as deleterious mutations, and the metabolic costs of maintaining systems to reduce the mutation rate, such as DNA repair enzymes. Viruses that use RNA as their genetic material have rapid mutation rates, which can be an advantage since these viruses will evolve constantly and rapidly, and thus evade the defensive responses of e.g. the human immune system.
Mutations can involve large sections of DNA becoming duplicated, usually through genetic recombination. These duplications are a major source of raw material for evolving new genes, with tens to hundreds of genes duplicated in animal genomes every million years. Most genes belong to larger families of genes of shared ancestry. Novel genes are produced by several methods, commonly through the duplication and mutation of an ancestral gene, or by recombining parts of different genes to form new combinations with new functions.
Here, domains act as modules, each with a particular and independent function, that can be mixed together to produce genes encoding new proteins with novel properties. For example, the human eye uses four genes to make structures that sense light: three for color vision and one for night vision; all four arose from a single ancestral gene. Another advantage of duplicating a gene (or even an entire genome) is that this increases redundancy; this allows one gene in the pair to acquire a new function while the other copy performs the original function. Other types of mutation occasionally create new genes from previously noncoding DNA.
### Selection
*Selection* is the process by which heritable traits that make it more likely for an organism to survive and successfully reproduce become more common in a population over successive generations.
It is sometimes valuable to distinguish between naturally occurring selection, natural selection, and selection that is a manifestation of choices made by humans, artificial selection. This distinction is rather diffuse. Natural selection is nevertheless the dominant part of selection. The natural genetic variation within a population of organisms means that some individuals will survive more successfully than others in their current environment. Factors which affect reproductive success are also important, an issue which Charles Darwin developed in his ideas on sexual selection.
Natural selection acts on the phenotype, or the observable characteristics of an organism, but the genetic (heritable) basis of any phenotype which gives a reproductive advantage will become more common in a population (see allele frequency). Over time, this process can result in adaptations that specialize organisms for particular ecological niches and may eventually result in the speciation (the emergence of new species).
Natural selection is one of the cornerstones of modern biology. The term was introduced by Darwin in his groundbreaking 1859 book *On the Origin of Species*, in which natural selection was described by analogy to artificial selection, a process by which animals and plants with traits considered desirable by human breeders are systematically favored for reproduction. The concept of natural selection was originally developed in the absence of a valid theory of heredity; at the time of Darwin\'s writing, nothing was known of modern genetics. The union of traditional Darwinian evolution with subsequent discoveries in classical and molecular genetics is termed the *modern evolutionary synthesis*. Natural selection remains the primary explanation for adaptive evolution.
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# Microevolution
## Four processes {#four_processes}
### Genetic drift {#genetic_drift}
Genetic drift is the change in the relative frequency in which a gene variant (allele) occurs in a population due to random sampling. That is, the alleles in the offspring in the population are a random sample of those in the parents. And chance has a role in determining whether a given individual survives and reproduces. A population\'s allele frequency is the fraction or percentage of its gene copies compared to the total number of gene alleles that share a particular form.
Genetic drift is an evolutionary process which leads to changes in allele frequencies over time. It may cause gene variants to disappear completely, and thereby reduce genetic variability. In contrast to natural selection, which makes gene variants more common or less common depending on their reproductive success, the changes due to genetic drift are not driven by environmental or adaptive pressures, and may be beneficial, neutral, or detrimental to reproductive success.
The effect of genetic drift is larger in small populations, and smaller in large populations. Vigorous debates wage among scientists over the relative importance of genetic drift compared with natural selection. Ronald Fisher held the view that genetic drift plays at the most a minor role in evolution, and this remained the dominant view for several decades. In 1968 Motoo Kimura rekindled the debate with his neutral theory of molecular evolution which claims that most of the changes in the genetic material are caused by genetic drift. The predictions of neutral theory, based on genetic drift, do not fit recent data on whole genomes well: these data suggest that the frequencies of neutral alleles change primarily due to selection at linked sites, rather than due to genetic drift by means of sampling error.
### Gene flow {#gene_flow}
Gene flow is the exchange of genes between populations, which are usually of the same species. Examples of gene flow within a species include the migration and then breeding of organisms, or the exchange of pollen. Gene transfer between species includes the formation of hybrid organisms and horizontal gene transfer.
Migration into or out of a population can change allele frequencies, as well as introducing genetic variation into a population. Immigration may add new genetic material to the established gene pool of a population. Conversely, emigration may remove genetic material. As barriers to reproduction between two diverging populations are required for the populations to become new species, gene flow may slow this process by spreading genetic differences between the populations. Gene flow is hindered by mountain ranges, oceans and deserts or even man-made structures such as the Great Wall of China, which has hindered the flow of plant genes.
Depending on how far two species have diverged since their most recent common ancestor, it may still be possible for them to produce offspring, as with horses and donkeys mating to produce mules. Such hybrids are generally infertile, due to the two different sets of chromosomes being unable to pair up during meiosis. In this case, closely related species may regularly interbreed, but hybrids will be selected against and the species will remain distinct. However, viable hybrids are occasionally formed and these new species can either have properties intermediate between their parent species, or possess a totally new phenotype. The importance of hybridization in developing new species of animals is unclear, although cases have been seen in many types of animals, with the gray tree frog being a particularly well-studied example.
Hybridization is, however, an important means of speciation in plants, since polyploidy (having more than two copies of each chromosome) is tolerated in plants more readily than in animals. Polyploidy is important in hybrids as it allows reproduction, with the two different sets of chromosomes each being able to pair with an identical partner during meiosis. Polyploid hybrids also have more genetic diversity, which allows them to avoid inbreeding depression in small populations.
Horizontal gene transfer is the transfer of genetic material from one organism to another organism that is not its offspring; this is most common among bacteria. In medicine, this contributes to the spread of antibiotic resistance, as when one bacteria acquires resistance genes it can rapidly transfer them to other species. Horizontal transfer of genes from bacteria to eukaryotes such as the yeast *Saccharomyces cerevisiae* and the adzuki bean beetle *Callosobruchus chinensis* may also have occurred. An example of larger-scale transfers are the eukaryotic bdelloid rotifers, which appear to have received a range of genes from bacteria, fungi, and plants. Viruses can also carry DNA between organisms, allowing transfer of genes even across biological domains. Large-scale gene transfer has also occurred between the ancestors of eukaryotic cells and prokaryotes, during the acquisition of chloroplasts and mitochondria.
*Gene flow* is the transfer of alleles from one population to another.
Migration into or out of a population may be responsible for a marked change in allele frequencies. Immigration may also result in the addition of new genetic variants to the established gene pool of a particular species or population.
There are a number of factors that affect the rate of gene flow between different populations. One of the most significant factors is mobility, as greater mobility of an individual tends to give it greater migratory potential. Animals tend to be more mobile than plants, although pollen and seeds may be carried great distances by animals or wind.
Maintained gene flow between two populations can also lead to a combination of the two gene pools, reducing the genetic variation between the two groups. It is for this reason that gene flow strongly acts against speciation, by recombining the gene pools of the groups, and thus, repairing the developing differences in genetic variation that would have led to full speciation and creation of daughter species.
For example, if a species of grass grows on both sides of a highway, pollen is likely to be transported from one side to the other and vice versa. If this pollen is able to fertilise the plant where it ends up and produce viable offspring, then the alleles in the pollen have effectively been able to move from the population on one side of the highway to the other.
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# Microevolution
## Origin and extended use of the term {#origin_and_extended_use_of_the_term}
### Origin
The term *microevolution* was first used by botanist Robert Greenleaf Leavitt in the journal *Botanical Gazette* in 1909, addressing what he called the \"mystery\" of how formlessness gives rise to form.
: *..The production of form from formlessness in the egg-derived individual, the multiplication of parts and the orderly creation of diversity among them, in an actual evolution, of which anyone may ascertain the facts, but of which no one has dissipated the mystery in any significant measure. This **microevolution** forms an integral part of the grand evolution problem and lies at the base of it, so that we shall have to understand the minor process before we can thoroughly comprehend the more general one\...*
However, Leavitt was using the term to describe what we would now call developmental biology; it was not until Russian Entomologist Yuri Filipchenko used the terms \"macroevolution\" and \"microevolution\" in 1927 in his German language work, *Variabilität und Variation*, that it attained its modern usage. The term was later brought into the English-speaking world by Filipchenko\'s student Theodosius Dobzhansky in his book Genetics and the Origin of Species (1937).
### Use in creationism {#use_in_creationism}
In young Earth creationism and baraminology a central tenet is that evolution can explain diversity in a limited number of created kinds which can interbreed (which they call \"microevolution\") while the formation of new \"kinds\" (which they call \"macroevolution\") is impossible. This acceptance of \"microevolution\" only within a \"kind\" is also typical of old Earth creationism.
Scientific organizations such as the American Association for the Advancement of Science describe microevolution as small scale change within species, and macroevolution as the formation of new species, but otherwise not being different from microevolution. In macroevolution, an accumulation of microevolutionary changes leads to speciation. The main difference between the two processes is that one occurs within a few generations, whilst the other takes place over thousands of years (i.e. a quantitative difference). Essentially they describe the same process; although evolution beyond the species level results in beginning and ending generations which could not interbreed, the intermediate generations could.
Opponents to creationism argue that changes in the number of chromosomes can be accounted for by intermediate stages in which a single chromosome divides in generational stages, or multiple chromosomes fuse, and cite the chromosome difference between humans and the other great apes as an example. Creationists insist that since the actual divergence between the other great apes and humans was not observed, the evidence is circumstantial.
Describing the fundamental similarity between macro and microevolution in his authoritative textbook \"Evolutionary Biology,\" biologist Douglas Futuyma writes,
Contrary to the claims of some antievolution proponents, evolution of life forms beyond the species level (i.e. speciation) has indeed been observed and documented by scientists on numerous occasions. In creation science, creationists accepted speciation as occurring within a \"created kind\" or \"baramin\", but objected to what they called \"third level-macroevolution\" of a new genus or higher rank in taxonomy. There is ambiguity in the ideas as to where to draw a line on \"species\", \"created kinds\", and what events and lineages fall within the rubric of microevolution or macroevolution
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# Multiple inheritance
**Multiple inheritance** is a feature of some object-oriented computer programming languages in which an object or class can inherit features from more than one parent object or parent class. It is distinct from single inheritance, where an object or class may only inherit from one particular object or class.
Multiple inheritance has been a controversial issue for many years, with opponents pointing to its increased complexity and ambiguity in situations such as the \"diamond problem\", where it may be ambiguous as to which parent class a particular feature is inherited from if more than one parent class implements said feature. This can be addressed in various ways, including using virtual inheritance. Alternate methods of object composition not based on inheritance such as mixins and traits have also been proposed to address the ambiguity.
## Details
In object-oriented programming (OOP), *inheritance* describes a relationship between two classes in which one class (the *child* class) *subclasses* the *parent* class. The child inherits methods and attributes of the parent, allowing for shared functionality. For example, one might create a variable class *Mammal* with features such as eating, reproducing, etc.; then define a child class *Cat* that inherits those features without having to explicitly program them, while adding new features like *chasing mice*.
Multiple inheritance allows programmers to use more than one totally orthogonal hierarchy simultaneously, such as allowing *Cat* to inherit from *Cartoon character* and *Pet* and *Mammal* and access features from within all of those classes.
## Implementations
Languages that support multiple inheritance include: C++, Common Lisp (via Common Lisp Object System (CLOS)), EuLisp (via The EuLisp Object System TELOS), Curl, Dylan, Eiffel, Logtalk, Object REXX, Scala (via use of mixin classes), OCaml, Perl, POP-11, Python, R, Raku, and Tcl (built-in from 8.6 or via Incremental Tcl (Incr Tcl) in earlier versions).
IBM System Object Model (SOM) runtime supports multiple inheritance, and any programming language targeting SOM can implement new SOM classes inherited from multiple bases.
Some object-oriented languages, such as Swift, Java, Fortran since its 2003 revision, C#, and Ruby implement *single inheritance*, although protocols, or *interfaces,* provide some of the functionality of true multiple inheritance.
PHP uses traits classes to inherit specific method implementations. Ruby uses modules to inherit multiple methods.
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# Multiple inheritance
## The diamond problem {#the_diamond_problem}
The \"**diamond problem**\" (sometimes referred to as the \"Deadly Diamond of Death\") is an ambiguity that arises when two classes B and C inherit from A, and class D inherits from both B and C. If there is a method in A that B and C have overridden, and D does not override it, then which version of the method does D inherit: that of B, or that of C?
For example, in the context of GUI software development, a class `Button` may inherit from both classes `Rectangle` (for appearance) and `Clickable` (for functionality/input handling), and classes `Rectangle` and `Clickable` both inherit from the `Object` class. Now if the `equals` method is called for a `Button` object and there is no such method in the `Button` class but there is an overridden `equals` method in `Rectangle` or `Clickable` (or both), which method should be eventually called?
It is called the \"diamond problem\" because of the shape of the class inheritance diagram in this situation. In this case, class A is at the top, both B and C separately beneath it, and D joins the two together at the bottom to form a diamond shape.
### Mitigation
Languages have different ways of dealing with these problems of repeated inheritance.
- C# (since C# 8.0) allows default interface method implementation, causing a class `A`, implementing interfaces `Ia` and `Ib` with similar methods having default implementations, to have two \"inherited\" methods with the same signature, causing the diamond problem. It is mitigated either by requiring `A` to implement the method itself, hence removing ambiguity, or forcing the caller to first cast the `A` object to the appropriate interface to use its default implementation of that method (e.g. `((Ia) aInstance).Method();`).
- C++ by default follows each inheritance path separately, so a `D` object would actually contain two separate `A` objects, and uses of `A`\'s members have to be properly qualified. If the inheritance from `A` to `B` and the inheritance from `A` to `C` are both marked \"`virtual`\" (for example, \"`class B : virtual public A`\"), C++ takes special care to only create one `A` object, and uses of `A`\'s members work correctly. If virtual inheritance and nonvirtual inheritance are mixed, there is a single virtual `A`, and a nonvirtual `A` for each nonvirtual inheritance path to `A`. C++ requires stating explicitly which parent class the feature to be used is invoked from i.e. `Worker::Human.Age`. C++ does not support explicit repeated inheritance since there would be no way to qualify which superclass to use (i.e. having a class appear more than once in a single derivation list \[class Dog : public Animal, Animal\]). C++ also allows a single instance of the multiple class to be created via the virtual inheritance mechanism (i.e. `Worker::Human` and `Musician::Human` will reference the same object).
- Common Lisp CLOS attempts to provide both reasonable default behavior and the ability to override it. By default, to put it simply, the methods are sorted in `D,B,C,A`, when B is written before C in the class definition. The method with the most specific argument classes is chosen (D\>(B,C)\>A) ; then in the order in which parent classes are named in the subclass definition (B\>C). However, the programmer can override this, by giving a specific method resolution order or stating a rule for combining methods. This is called method combination, which may be fully controlled. The MOP (metaobject protocol) also provides means to modify the inheritance, dynamic dispatch, class instantiation, and other internal mechanisms without affecting the stability of the system.
- Curl allows only classes that are explicitly marked as *shared* to be inherited repeatedly. Shared classes must define a *secondary constructor* for each regular constructor in the class. The regular constructor is called the first time the state for the shared class is initialized through a subclass constructor, and the secondary constructor will be invoked for all other subclasses.
- In Eiffel, the ancestors\' features are chosen explicitly with select and rename directives. This allows the features of the base class to be shared between its descendants or to give each of them a separate copy of the base class. Eiffel allows explicit joining or separation of features inherited from ancestor classes. Eiffel will automatically join features together, if they have the same name and implementation. The class writer has the option to rename the inherited features to separate them. Multiple inheritance is a frequent occurrence in Eiffel development; most of the effective classes in the widely used EiffelBase library of data structures and algorithms, for example, have two or more parents.
- Go prevents the diamond problem at compile time. If a structure `D` embeds two structures `B` and `C` which both have a method `F()`, thus satisfying an interface `A`, the compiler will complain about an \"ambiguous selector\" if `D.F()` is called, or if an instance of `D` is assigned to a variable of type `A`. `B` and `C`\'s methods can be called explicitly with `D.B.F()` or `D.C.F()`.
- Java 8 introduces default methods on interfaces. If `A,B,C` are interfaces, `B,C` can each provide a different implementation to an abstract method of `A`, causing the diamond problem. Either class `D` must reimplement the method (the body of which can simply forward the call to one of the super implementations), or the ambiguity will be rejected as a compile error. Prior to Java 8, Java was not subject to the diamond problem risk, because it did not support multiple inheritance and interface default methods were not available.
- JavaFX Script in version 1.2 allows multiple inheritance through the use of mixins. In case of conflict, the compiler prohibits the direct usage of the ambiguous variable or function. Each inherited member can still be accessed by casting the object to the mixin of interest, e.g. `(individual as Person).printInfo();`.
- Kotlin allows multiple inheritance of Interfaces, however, in a Diamond problem scenario, the child class must override the method that causes the inheritance conflict and specify which parent class implementation should be used. eg `super``.someMethod()`
- Logtalk supports both interface and implementation multi-inheritance, allowing the declaration of method *aliases* that provide both renaming and access to methods that would be masked out by the default conflict resolution mechanism.
- In OCaml, parent classes are specified individually in the body of the class definition. Methods (and attributes) are inherited in the same order, with each newly inherited method overriding any existing methods. OCaml chooses the last matching definition of a class inheritance list to resolve which method implementation to use under ambiguities. To override the default behavior, one simply qualifies a method call with the desired class definition.
- Perl uses the list of classes to inherit from as an ordered list. The compiler uses the first method it finds by depth-first searching of the superclass list or using the C3 linearization of the class hierarchy. Various extensions provide alternative class composition schemes. The order of inheritance affects the class semantics. In the above ambiguity, class `B` and its ancestors would be checked before class `C` and its ancestors, so the method in `A` would be inherited through `B`. This is shared with Io and Picolisp. In Perl, this behavior can be overridden using the `mro` or other modules to use C3 linearization or other algorithms.
- Python has the same structure as Perl, but, unlike Perl, includes it in the syntax of the language. The order of inheritance affects the class semantics. Python had to deal with this upon the introduction of new-style classes, all of which have a common ancestor, `object`. Python creates a list of classes using the C3 linearization (or Method Resolution Order (MRO)) algorithm. That algorithm enforces two constraints: children precede their parents and if a class inherits from multiple classes, they are kept in the order specified in the tuple of base classes (however in this case, some classes high in the inheritance graph may precede classes lower in the graph). Thus, the method resolution order is: `D`, `B`, `C`, `A`.
- Ruby classes have exactly one parent but may also inherit from multiple *modules;* ruby class definitions are executed, and the (re)definition of a method obscures any previously existing definition at the time of execution. In the absence of runtime metaprogramming this has approximately the same semantics as rightmost depth first resolution.
- Scala allows multiple instantiation of *traits*, which allows for multiple inheritance by adding a distinction between the class hierarchy and the trait hierarchy. A class can only inherit from a single class, but can mix-in as many traits as desired. Scala resolves method names using a right-first depth-first search of extended \'traits\', before eliminating all but the last occurrence of each module in the resulting list. So, the resolution order is: \[`D`, `C`, `A`, `B`, `A`\], which reduces down to \[`D`, `C`, `B`, `A`\].
- Tcl allows multiple parent classes; the order of specification in the class declaration affects the name resolution for members using the C3 linearization algorithm.
Languages that allow only single inheritance, where a class can only derive from one base class, do not have the diamond problem. The reason for this is that such languages have at most one implementation of any method at any level in the inheritance chain regardless of the repetition or placement of methods. Typically these languages allow classes to implement multiple protocols, called interfaces in Java. These protocols define methods but do not provide concrete implementations. This strategy has been used by ActionScript, C#, D, Java, Nemerle, Object Pascal, Objective-C, Smalltalk, Swift and PHP. All these languages allow classes to implement multiple protocols.
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# Multiple inheritance
## The diamond problem {#the_diamond_problem}
### Mitigation
Moreover, Ada, C#, Java, Object Pascal, Objective-C, Swift and PHP allow multiple-inheritance of interfaces (called protocols in Objective-C and Swift). Interfaces are like abstract base classes that specify method signatures without implementing any behaviour. (\"Pure\" interfaces such as the ones in Java up to version 7 do not permit any implementation or instance data in the interface.) Nevertheless, even when several interfaces declare the same method signature, as soon as that method is implemented (defined) anywhere in the inheritance chain, it overrides any implementation of that method in the chain above it (in its superclasses). Hence, at any given level in the inheritance chain, there can be at most one implementation of any method. Thus, single-inheritance method implementation does not exhibit the diamond problem even with multiple-inheritance of interfaces. With the introduction of default implementation for interfaces in Java 8 and C# 8, it is still possible to generate a diamond problem, although this will only appear as a compile-time error
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# Molecule
A **molecule** is a group of two or more atoms that are held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemistry, and biochemistry, the distinction from ions is dropped and *molecule* is often used when referring to polyatomic ions.
A molecule may be homonuclear, that is, it consists of atoms of one chemical element, e.g. two atoms in the oxygen molecule (O~2~); or it may be heteronuclear, a chemical compound composed of more than one element, e.g. water (two hydrogen atoms and one oxygen atom; H~2~O). In the kinetic theory of gases, the term *molecule* is often used for any gaseous particle regardless of its composition. This relaxes the requirement that a molecule contains two or more atoms, since the noble gases are individual atoms. Atoms and complexes connected by non-covalent interactions, such as hydrogen bonds or ionic bonds, are typically not considered single molecules.
Concepts similar to molecules have been discussed since ancient times, but modern investigation into the nature of molecules and their bonds began in the 17th century. Refined over time by scientists such as Robert Boyle, Amedeo Avogadro, Jean Perrin, and Linus Pauling, the study of molecules is today known as molecular physics or molecular chemistry.
## Etymology
According to Merriam-Webster and the Online Etymology Dictionary, the word \"molecule\" derives from the Latin \"moles\" or small unit of mass. The word is derived from French *`{{linktext|molécule}}`{=mediawiki}* (1678), from Neo-Latin *`{{linktext|molecula}}`{=mediawiki}*, diminutive of Latin *`{{linktext|moles}}`{=mediawiki}* \"mass, barrier\". The word, which until the late 18th century was used only in Latin form, became popular after being used in works of philosophy by Descartes.
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# Molecule
## History
The definition of the molecule has evolved as knowledge of the structure of molecules has increased. Earlier definitions were less precise, defining molecules as the smallest particles of pure chemical substances that still retain their composition and chemical properties. This definition often breaks down since many substances in ordinary experience, such as rocks, salts, and metals, are composed of large crystalline networks of chemically bonded atoms or ions, but are not made of discrete molecules.
The modern concept of molecules can be traced back towards pre-scientific and Greek philosophers such as Leucippus and Democritus who argued that all the universe is composed of atoms and voids. Circa 450 BC Empedocles imagined fundamental elements (fire (), earth (), air (), and water ()) and \"forces\" of attraction and repulsion allowing the elements to interact.
A fifth element, the incorruptible quintessence aether, was considered to be the fundamental building block of the heavenly bodies. The viewpoint of Leucippus and Empedocles, along with the aether, was accepted by Aristotle and passed to medieval and renaissance Europe.
In a more concrete manner, however, the concept of aggregates or units of bonded atoms, i.e. \"molecules\", traces its origins to Robert Boyle\'s 1661 hypothesis, in his famous treatise *The Sceptical Chymist*, that matter is composed of *clusters of particles* and that chemical change results from the rearrangement of the clusters. Boyle argued that matter\'s basic elements consisted of various sorts and sizes of particles, called \"corpuscles\", which were capable of arranging themselves into groups. In 1789, William Higgins published views on what he called combinations of \"ultimate\" particles, which foreshadowed the concept of valency bonds. If, for example, according to Higgins, the force between the ultimate particle of oxygen and the ultimate particle of nitrogen were 6, then the strength of the force would be divided accordingly, and similarly for the other combinations of ultimate particles.
Amedeo Avogadro created the word \"molecule\". His 1811 paper \"Essay on Determining the Relative Masses of the Elementary Molecules of Bodies\", he essentially states, i.e. according to Partington\'s *A Short History of Chemistry*, that:`{{Blockquote|The smallest particles of gases are not necessarily simple atoms, but are made up of a certain number of these atoms united by attraction to form a single '''molecule'''.}}`{=mediawiki}In coordination with these concepts, in 1833 the French chemist Marc Antoine Auguste Gaudin presented a clear account of Avogadro\'s hypothesis, regarding atomic weights, by making use of \"volume diagrams\", which clearly show both semi-correct molecular geometries, such as a linear water molecule, and correct molecular formulas, such as H~2~O: In 1917, an unknown American undergraduate chemical engineer named Linus Pauling was learning the Dalton hook-and-eye bonding method, which was the mainstream description of bonds between atoms at the time. Pauling, however, was not satisfied with this method and looked to the newly emerging field of quantum physics for a new method. In 1926, French physicist Jean Perrin received the Nobel Prize in physics for proving, conclusively, the existence of molecules. He did this by calculating the Avogadro constant using three different methods, all involving liquid phase systems. First, he used a gamboge soap-like emulsion, second by doing experimental work on Brownian motion, and third by confirming Einstein\'s theory of particle rotation in the liquid phase.
In 1927, the physicists Fritz London and Walter Heitler applied the new quantum mechanics to the deal with the saturable, nondynamic forces of attraction and repulsion, i.e., exchange forces, of the hydrogen molecule. Their valence bond treatment of this problem, in their joint paper, was a landmark in that it brought chemistry under quantum mechanics. Their work was an influence on Pauling, who had just received his doctorate and visited Heitler and London in Zürich on a Guggenheim Fellowship.
Subsequently, in 1931, building on the work of Heitler and London and on theories found in Lewis\' famous article, Pauling published his ground-breaking article \"The Nature of the Chemical Bond\" in which he used quantum mechanics to calculate properties and structures of molecules, such as angles between bonds and rotation about bonds. On these concepts, Pauling developed hybridization theory to account for bonds in molecules such as CH~4~, in which four sp³ hybridised orbitals are overlapped by hydrogen\'s *1s* orbital, yielding four sigma (σ) bonds. The four bonds are of the same length and strength, which yields a molecular structure as shown below:
## Molecular science {#molecular_science}
The science of molecules is called *molecular chemistry* or *molecular physics*, depending on whether the focus is on chemistry or physics. Molecular chemistry deals with the laws governing the interaction between molecules that results in the formation and breakage of chemical bonds, while molecular physics deals with the laws governing their structure and properties. In practice, however, this distinction is vague. In molecular sciences, a molecule consists of a stable system (bound state) composed of two or more atoms. Polyatomic ions may sometimes be usefully thought of as electrically charged molecules. The term *unstable molecule* is used for very reactive species, i.e., short-lived assemblies (resonances) of electrons and nuclei, such as radicals, molecular ions, Rydberg molecules, transition states, van der Waals complexes, or systems of colliding atoms as in Bose--Einstein condensate.
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# Molecule
## Prevalence
Molecules as components of matter are common. They also make up most of the oceans and atmosphere. Most organic substances are molecules. The substances of life are molecules, e.g. proteins, the amino acids of which they are composed, the nucleic acids (DNA and RNA), sugars, carbohydrates, fats, and vitamins. The nutrient minerals are generally ionic compounds, thus they are not molecules, e.g. iron sulfate.
However, the majority of familiar solid substances on Earth are made partly or completely of crystals or ionic compounds, which are not made of molecules. These include all of the minerals that make up the substance of the Earth, sand, clay, pebbles, rocks, boulders, bedrock, the molten interior, and the core of the Earth. All of these contain many chemical bonds, but are *not* made of identifiable molecules.
No typical molecule can be defined for salts nor for covalent crystals, although these are often composed of repeating unit cells that extend either in a plane, e.g. graphene; or three-dimensionally e.g. diamond, quartz, sodium chloride. The theme of repeated unit-cellular-structure also holds for most metals which are condensed phases with metallic bonding. Thus solid metals are not made of molecules. In glasses, which are solids that exist in a vitreous disordered state, the atoms are held together by chemical bonds with no presence of any definable molecule, nor any of the regularity of repeating unit-cellular-structure that characterizes salts, covalent crystals, and metals.
## Bonding
Molecules are generally held together by covalent bonding. Several non-metallic elements exist only as molecules in the environment either in compounds or as homonuclear molecules, not as free atoms: for example, hydrogen.
While some people say a metallic crystal can be considered a single giant molecule held together by metallic bonding, others point out that metals behave very differently than molecules.
### Covalent
*Main article: Covalent bonding* A covalent bond is a chemical bond that involves the sharing of electron pairs between atoms. These electron pairs are termed *shared pairs* or *bonding pairs*, and the stable balance of attractive and repulsive forces between atoms, when they share electrons, is termed *covalent bonding*.
### Ionic
Ionic bonding is a type of chemical bond that involves the electrostatic attraction between oppositely charged ions, and is the primary interaction occurring in ionic compounds. The ions are atoms that have lost one or more electrons (termed cations) and atoms that have gained one or more electrons (termed anions). This transfer of electrons is termed *electrovalence* in contrast to covalence. In the simplest case, the cation is a metal atom and the anion is a nonmetal atom, but these ions can be of a more complicated nature, e.g. molecular ions like NH~4~^+^ or SO~4~^2−^. At normal temperatures and pressures, ionic bonding mostly creates solids (or occasionally liquids) without separate identifiable molecules, but the vaporization/sublimation of such materials does produce separate molecules where electrons are still transferred fully enough for the bonds to be considered ionic rather than covalent.
## Molecular size {#molecular_size}
Most molecules are far too small to be seen with the naked eye, although molecules of many polymers can reach macroscopic sizes, including biopolymers such as DNA. Molecules commonly used as building blocks for organic synthesis have a dimension of a few angstroms (Å) to several dozen Å, or around one billionth of a meter. Single molecules cannot usually be observed by light (as noted above), but small molecules and even the outlines of individual atoms may be traced in some circumstances by use of an atomic force microscope. Some of the largest molecules are macromolecules or supermolecules.
The smallest molecule is the diatomic hydrogen (H~2~), with a bond length of 0.74 Å.
Effective molecular radius is the size a molecule displays in solution. The table of permselectivity for different substances contains examples.
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# Molecule
## Molecular formulas {#molecular_formulas}
### Chemical formula types {#chemical_formula_types}
The chemical formula for a molecule uses one line of chemical element symbols, numbers, and sometimes also other symbols, such as parentheses, dashes, brackets, and *plus* (+) and *minus* (−) signs. These are limited to one typographic line of symbols, which may include subscripts and superscripts.
A compound\'s empirical formula is a very simple type of chemical formula. It is the simplest integer ratio of the chemical elements that constitute it. For example, water is always composed of a 2:1 ratio of hydrogen to oxygen atoms, and ethanol (ethyl alcohol) is always composed of carbon, hydrogen, and oxygen in a 2:6:1 ratio. However, this does not determine the kind of molecule uniquely -- dimethyl ether has the same ratios as ethanol, for instance. Molecules with the same atoms in different arrangements are called isomers. Also carbohydrates, for example, have the same ratio (carbon:hydrogen:oxygen= 1:2:1) (and thus the same empirical formula) but different total numbers of atoms in the molecule.
The molecular formula reflects the exact number of atoms that compose the molecule and so characterizes different molecules. However different isomers can have the same atomic composition while being different molecules.
The empirical formula is often the same as the molecular formula but not always. For example, the molecule acetylene has molecular formula C~2~H~2~, but the simplest integer ratio of elements is CH.
The molecular mass can be calculated from the chemical formula and is typically expressed in daltons, which are equal to 1/12 of the mass of a neutral carbon-12 (^12^C isotope) atom. For network solids, the term formula unit is used in stoichiometric calculations.
### Structural formula {#structural_formula}
thumb\|right\|upright=1.8\|3D (left and center) and 2D (right) representations of the terpenoid molecule atisane *Main article: Structural formula*
For molecules with a complicated 3-dimensional structure, especially involving atoms bonded to four different substituents, a simple molecular formula or even semi-structural chemical formula may not be enough to completely specify the molecule. In this case, a graphical type of formula called a structural formula may be needed. Structural formulas may in turn be represented with a one-dimensional chemical name, but such chemical nomenclature requires many words and terms which are not part of chemical formulas.
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# Molecule
## Molecular geometry {#molecular_geometry}
Molecules have fixed equilibrium geometries---bond lengths and angles--- about which they continuously oscillate through vibrational and rotational motions. A pure substance is composed of molecules with the same average geometrical structure. The chemical formula and the structure of a molecule are the two important factors that determine its properties, particularly its reactivity. Isomers share a chemical formula but normally have very different properties because of their different structures. Stereoisomers, a particular type of isomer, may have very similar physico-chemical properties and at the same time different biochemical activities.
## Molecular spectroscopy {#molecular_spectroscopy}
thumb\|upright=1.3\|Hydrogen can be removed from individual H~2~TPP molecules by applying excess voltage to the tip of a scanning tunneling microscope (STM, a); this removal alters the current-voltage (I-V) curves of TPP molecules, measured using the same STM tip, from diode like (red curve in b) to resistor like (green curve). Image (c) shows a row of TPP, H~2~TPP and TPP molecules. While scanning image (d), excess voltage was applied to H~2~TPP at the black dot, which instantly removed hydrogen, as shown in the bottom part of (d) and in the rescan image (e). Such manipulations can be used in single-molecule electronics.
**Molecular spectroscopy** deals with the response (spectrum) of molecules interacting with probing signals of known energy (or frequency, according to the Planck relation). Molecules have quantized energy levels that can be analyzed by detecting the molecule\'s energy exchange through absorbance or emission. Spectroscopy does not generally refer to diffraction studies where particles such as neutrons, electrons, or high energy X-rays interact with a regular arrangement of molecules (as in a crystal).
Microwave spectroscopy commonly measures changes in the rotation of molecules, and can be used to identify molecules in outer space. Infrared spectroscopy measures the vibration of molecules, including stretching, bending or twisting motions. It is commonly used to identify the kinds of bonds or functional groups in molecules. Changes in the arrangements of electrons yield absorption or emission lines in ultraviolet, visible or near infrared light, and result in colour. Nuclear resonance spectroscopy measures the environment of particular nuclei in the molecule, and can be used to characterise the numbers of atoms in different positions in a molecule.
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# Molecule
## Theoretical aspects {#theoretical_aspects}
The study of molecules by molecular physics and theoretical chemistry is largely based on quantum mechanics and is essential for the understanding of the chemical bond. The simplest of molecules is the hydrogen molecule-ion, H~2~^+^, and the simplest of all the chemical bonds is the one-electron bond. H~2~^+^ is composed of two positively charged protons and one negatively charged electron, which means that the Schrödinger equation for the system can be solved more easily due to the lack of electron--electron repulsion. With the development of fast digital computers, approximate solutions for more complicated molecules became possible and are one of the main aspects of computational chemistry.
When trying to define rigorously whether an arrangement of atoms is *sufficiently stable* to be considered a molecule, IUPAC suggests that it \"must correspond to a depression on the potential energy surface that is deep enough to confine at least one vibrational state\". This definition does not depend on the nature of the interaction between the atoms, but only on the strength of the interaction. In fact, it includes weakly bound species that would not traditionally be considered molecules, such as the helium dimer, He~2~, which has one vibrational bound state and is so loosely bound that it is only likely to be observed at very low temperatures.
Whether or not an arrangement of atoms is *sufficiently stable* to be considered a molecule is inherently an operational definition. Philosophically, therefore, a molecule is not a fundamental entity (in contrast, for instance, to an elementary particle); rather, the concept of a molecule is the chemist\'s way of making a useful statement about the strengths of atomic-scale interactions in the world that we observe
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# Mechanics
`{{Classical mechanics |branches}}`{=mediawiki} `{{Quantum mechanics |background}}`{=mediawiki}
**Mechanics** (`{{etymology|grc|''{{wikt-lang|grc|μηχανική}}'' ({{grc-transl|μηχανική}})|of [[machine]]s}}`{=mediawiki}) is the area of physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects may result in displacements, which are changes of an object\'s position relative to its environment.
Theoretical expositions of this branch of physics has its origins in Ancient Greece, for instance, in the writings of Aristotle and Archimedes (see History of classical mechanics and Timeline of classical mechanics). During the early modern period, scientists such as Galileo Galilei, Johannes Kepler, Christiaan Huygens, and Isaac Newton laid the foundation for what is now known as classical mechanics.
As a branch of classical physics, mechanics deals with bodies that are either at rest or are moving with velocities significantly less than the speed of light. It can also be defined as the physical science that deals with the motion of and forces on bodies not in the quantum realm.
## History
### Antiquity
The ancient Greek philosophers were among the first to propose that abstract principles govern nature. The main theory of mechanics in antiquity was Aristotelian mechanics, though an alternative theory is exposed in the pseudo-Aristotelian *Mechanical Problems*, often attributed to one of his successors.
There is another tradition that goes back to the ancient Greeks where mathematics is used more extensively to analyze bodies statically or dynamically, an approach that may have been stimulated by prior work of the Pythagorean Archytas. Examples of this tradition include pseudo-Euclid (*On the Balance*), Archimedes (*On the Equilibrium of Planes*, *On Floating Bodies*), Hero (*Mechanica*), and Pappus (*Collection*, Book VIII).
### Medieval age {#medieval_age}
In the Middle Ages, Aristotle\'s theories were criticized and modified by a number of figures, beginning with John Philoponus in the 6th century. A central problem was that of projectile motion, which was discussed by Hipparchus and Philoponus.
Persian Islamic polymath Ibn Sīnā published his theory of motion in *The Book of Healing* (1020). He said that an impetus is imparted to a projectile by the thrower, and viewed it as persistent, requiring external forces such as air resistance to dissipate it. Ibn Sina made distinction between \'force\' and \'inclination\' (called \"mayl\"), and argued that an object gained mayl when the object is in opposition to its natural motion. So he concluded that continuation of motion is attributed to the inclination that is transferred to the object, and that object will be in motion until the mayl is spent. He also claimed that a projectile in a vacuum would not stop unless it is acted upon, consistent with Newton\'s first law of motion.
On the question of a body subject to a constant (uniform) force, the 12th-century Jewish-Arab scholar Hibat Allah Abu\'l-Barakat al-Baghdaadi (born Nathanel, Iraqi, of Baghdad) stated that constant force imparts constant acceleration. According to Shlomo Pines, al-Baghdaadi\'s theory of motion was \"the oldest negation of Aristotle\'s fundamental dynamic law \[namely, that a constant force produces a uniform motion\], \[and is thus an\] anticipation in a vague fashion of the fundamental law of classical mechanics \[namely, that a force applied continuously produces acceleration\].\"
Influenced by earlier writers such as Ibn Sina and al-Baghdaadi, the 14th-century French priest Jean Buridan developed the theory of impetus, which later developed into the modern theories of inertia, velocity, acceleration and momentum. This work and others was developed in 14th-century England by the Oxford Calculators such as Thomas Bradwardine, who studied and formulated various laws regarding falling bodies. The concept that the main properties of a body are uniformly accelerated motion (as of falling bodies) was worked out by the 14th-century Oxford Calculators.
### Early modern age {#early_modern_age}
Two central figures in the early modern age are Galileo Galilei and Isaac Newton. Galileo\'s final statement of his mechanics, particularly of falling bodies, is his *Two New Sciences* (1638). Newton\'s 1687 *Philosophiæ Naturalis Principia Mathematica* provided a detailed mathematical account of mechanics, using the newly developed mathematics of calculus and providing the basis of Newtonian mechanics.
There is some dispute over priority of various ideas: Newton\'s *Principia* is certainly the seminal work and has been tremendously influential, and many of the mathematics results therein could not have been stated earlier without the development of the calculus. However, many of the ideas, particularly as pertain to inertia and falling bodies, had been developed by prior scholars such as Christiaan Huygens and the less-known medieval predecessors. Precise credit is at times difficult or contentious because scientific language and standards of proof changed, so whether medieval statements are *equivalent* to modern statements or *sufficient* proof, or instead *similar* to modern statements and *hypotheses* is often debatable.
### Modern age {#modern_age}
Two main modern developments in mechanics are general relativity of Einstein, and quantum mechanics, both developed in the 20th century based in part on earlier 19th-century ideas. The development in the modern continuum mechanics, particularly in the areas of elasticity, plasticity, fluid dynamics, electrodynamics, and thermodynamics of deformable media, started in the second half of the 20th century.
## Types of mechanical bodies {#types_of_mechanical_bodies}
The often-used term **body** needs to stand for a wide assortment of objects, including particles, projectiles, spacecraft, stars, parts of machinery, parts of solids, parts of fluids (gases and liquids), etc.
Other distinctions between the various sub-disciplines of mechanics concern the nature of the bodies being described. Particles are bodies with little (known) internal structure, treated as mathematical points in classical mechanics. Rigid bodies have size and shape, but retain a simplicity close to that of the particle, adding just a few so-called degrees of freedom, such as orientation in space.
Otherwise, bodies may be semi-rigid, i.e. elastic, or non-rigid, i.e. fluid. These subjects have both classical and quantum divisions of study.
For instance, the motion of a spacecraft, regarding its orbit and attitude (rotation), is described by the relativistic theory of classical mechanics, while the analogous movements of an atomic nucleus are described by quantum mechanics.
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# Mechanics
## Sub-disciplines {#sub_disciplines}
The following are the three main designations consisting of various subjects that are studied in mechanics.
Note that there is also the \"theory of fields\" which constitutes a separate discipline in physics, formally treated as distinct from mechanics, whether it be classical fields or quantum fields. But in actual practice, subjects belonging to mechanics and fields are closely interwoven. Thus, for instance, forces that act on particles are frequently derived from fields (electromagnetic or gravitational), and particles generate fields by acting as sources. In fact, in quantum mechanics, particles themselves are fields, as described theoretically by the wave function.
### Classical
The following are described as forming classical mechanics:
- Newtonian mechanics, the original theory of motion (kinematics) and forces (dynamics)
- Analytical mechanics is a reformulation of Newtonian mechanics with an emphasis on system energy, rather than on forces. There are two main branches of analytical mechanics:
- Hamiltonian mechanics, a theoretical formalism, based on the principle of conservation of energy
- Lagrangian mechanics, another theoretical formalism, based on the principle of the least action
- Classical statistical mechanics generalizes ordinary classical mechanics to consider systems in an unknown state; often used to derive thermodynamic properties.
- Celestial mechanics, the motion of bodies in space: planets, comets, stars, galaxies, etc.
- Astrodynamics, spacecraft navigation, etc.
- Solid mechanics, elasticity, plasticity, or viscoelasticity exhibited by deformable solids
- Fracture mechanics
- Acoustics, sound (density, variation, propagation) in solids, fluids and gases
- Statics, semi-rigid bodies in mechanical equilibrium
- Fluid mechanics, the motion of fluids
- Soil mechanics, mechanical behavior of soils
- Continuum mechanics, mechanics of continua (both solid and fluid)
- Hydraulics, mechanical properties of liquids
- Fluid statics, liquids in equilibrium
- Applied mechanics (also known as engineering mechanics)
- Biomechanics, solids, fluids, etc. in biology
- Biophysics, physical processes in living organisms
- Relativistic or Einsteinian mechanics
### Quantum
The following are categorized as being part of quantum mechanics:
- Schrödinger wave mechanics, used to describe the movements of the wavefunction of a single particle.
- Matrix mechanics is an alternative formulation that allows considering systems with a finite-dimensional state space.
- Quantum statistical mechanics generalizes ordinary quantum mechanics to consider systems in an unknown state; often used to derive thermodynamic properties.
- Particle physics, the motion, structure, and behavior of fundamental particles
- Nuclear physics, the motion, structure, and reactions of nuclei
- Condensed matter physics, quantum gases, solids, liquids, etc.
Historically, classical mechanics had been around for nearly a quarter millennium before quantum mechanics developed. Classical mechanics originated with Isaac Newton\'s laws of motion in Philosophiæ Naturalis Principia Mathematica, developed over the seventeenth century. Quantum mechanics developed later, over the nineteenth century, precipitated by Planck\'s postulate and Albert Einstein\'s explanation of the photoelectric effect. Both fields are commonly held to constitute the most certain knowledge that exists about physical nature.
Classical mechanics has especially often been viewed as a model for other so-called exact sciences. Essential in this respect is the extensive use of mathematics in theories, as well as the decisive role played by experiment in generating and testing them.
Quantum mechanics is of a bigger scope, as it encompasses classical mechanics as a sub-discipline which applies under certain restricted circumstances. According to the correspondence principle, there is no contradiction or conflict between the two subjects, each simply pertains to specific situations. The correspondence principle states that the behavior of systems described by quantum theories reproduces classical physics in the limit of large quantum numbers, i.e. if quantum mechanics is applied to large systems (for e.g. a baseball), the result would almost be the same if classical mechanics had been applied. Quantum mechanics has superseded classical mechanics at the foundation level and is indispensable for the explanation and prediction of processes at the molecular, atomic, and sub-atomic level. However, for macroscopic processes classical mechanics is able to solve problems which are unmanageably difficult (mainly due to computational limits) in quantum mechanics and hence remains useful and well used. Modern descriptions of such behavior begin with a careful definition of such quantities as displacement (distance moved), time, velocity, acceleration, mass, and force. Until about 400 years ago, however, motion was explained from a very different point of view. For example, following the ideas of Greek philosopher and scientist Aristotle, scientists reasoned that a cannonball falls down because its natural position is in the Earth; the Sun, the Moon, and the stars travel in circles around the Earth because it is the nature of heavenly objects to travel in perfect circles.
Often cited as father to modern science, Galileo brought together the ideas of other great thinkers of his time and began to calculate motion in terms of distance travelled from some starting position and the time that it took. He showed that the speed of falling objects increases steadily during the time of their fall. This acceleration is the same for heavy objects as for light ones, provided air friction (air resistance) is discounted. The English mathematician and physicist Isaac Newton improved this analysis by defining force and mass and relating these to acceleration. For objects traveling at speeds close to the speed of light, Newton\'s laws were superseded by Albert Einstein\'s theory of relativity. \[A sentence illustrating the computational complication of Einstein\'s theory of relativity.\] For atomic and subatomic particles, Newton\'s laws were superseded by quantum theory. For everyday phenomena, however, Newton\'s three laws of motion remain the cornerstone of dynamics, which is the study of what causes motion.
### Relativistic
Akin to the distinction between quantum and classical mechanics, Albert Einstein\'s general and special theories of relativity have expanded the scope of Newton and Galileo\'s formulation of mechanics. The differences between relativistic and Newtonian mechanics become significant and even dominant as the velocity of a body approaches the speed of light. For instance, in Newtonian mechanics, the kinetic energy of a free particle is `{{math|1=''E'' = {{sfrac|1|2}}''mv''<sup>2</sup>}}`{=mediawiki}, whereas in relativistic mechanics, it is `{{math|1=''E'' = (''γ'' − 1)''mc''<sup>2</sup>}}`{=mediawiki} (where `{{math|''γ''}}`{=mediawiki} is the Lorentz factor; this formula reduces to the Newtonian expression in the low energy limit).
For high-energy processes, quantum mechanics must be adjusted to account for special relativity; this has led to the development of quantum field theory
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# Main-group element
In chemistry and atomic physics, the **main group** is the group of elements (sometimes called the **representative elements**) whose lightest members are represented by helium, lithium, beryllium, boron, carbon, nitrogen, oxygen, and fluorine as arranged in the periodic table of the elements. The main group includes the elements (except hydrogen, which is sometimes not included) in groups 1 and 2 (s-block), and groups 13 to 18 (p-block). The s-block elements are primarily characterised by one main oxidation state, and the p-block elements, when they have multiple oxidation states, often have common oxidation states separated by two units.
Main-group elements (with some of the lighter transition metals) are the most abundant elements on Earth, in the Solar System, and in the universe. Group 12 elements are often considered to be transition metals; however, zinc (Zn), cadmium (Cd), and mercury (Hg) share some properties of both groups, and some scientists believe they should be included in the main group.
Occasionally, even the group 3 elements as well as the lanthanides and actinides have been included, because especially the group 3 elements and many lanthanides are electropositive elements with only one main oxidation state like the group 1 and 2 elements. The position of the actinides is more questionable, but the most common and stable of them, thorium (Th) and uranium (U), are similar to main-group elements as thorium is an electropositive element with only one main oxidation state (+4), and uranium has two main ones separated by two oxidation units (+4 and +6).
In older nomenclature, the main-group elements are groups IA and IIA, and groups IIIB to 0 (CAS groups IIIA to VIIIA). Group 12 is labelled as group IIB in both systems. Group 3 is labelled as group IIIA in the older nomenclature (CAS group IIIB)
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# Midrash
***Midrash*** (`{{IPAc-en|ˈ|m|ɪ|d|r|ɑː|ʃ}}`{=mediawiki}; *מִדְרָשׁ\]\]*; `{{abbr|pl.|plural}}`{=mediawiki} *מִדְרָשִׁים* `{{transliteration|he|midrashim}}`{=mediawiki} or `{{Script/Hebrew|מִדְרָשׁוֹת}}`{=mediawiki} *midrashot*) is an expansive Jewish Biblical exegesis using a rabbinic mode of interpretation prominent in the Talmud. The word itself means \"textual interpretation\", \"study\", or \"exegesis\", derived from the root verb `{{transliteration|he|darash}}`{=mediawiki} (*דָּרַשׁ*), which means \"resort to, seek, seek with care, enquire, require\".
Midrash and rabbinic readings \"discern value in texts, words, and letters, as potential revelatory spaces\", writes the Hebrew scholar Wilda Gafney. \"They reimagine dominant narratival readings while crafting new ones to stand alongside---not replace---former readings. Midrash also asks questions of the text; sometimes it provides answers, sometimes it leaves the reader to answer the questions\". Vanessa Lovelace defines midrash as \"a Jewish mode of interpretation that not only engages the words of the text, behind the text, and beyond the text, but also focuses on each letter, and the words left unsaid by each line\".
An example of a midrashic interpretation:
The term Midrash is also used of a rabbinic work that interprets Scripture in that manner. Such works contain early interpretations and commentaries on the Written Torah and Oral Torah (spoken law and sermons), as well as non-legalistic rabbinic literature (`{{transliteration|he|[[aggadah]]}}`{=mediawiki}) and occasionally Jewish religious laws (`{{transliteration|he|[[halakha]]}}`{=mediawiki}), which usually form a running commentary on specific passages in the Hebrew Scripture (Tanakh).
The word *Midrash*, especially if capitalized, can refer to a specific compilation of these rabbinic writings composed between 400 and 1200 CE. According to Gary Porton and Jacob Neusner, *midrash* has three technical meanings:
1. Judaic biblical interpretation;
2. the method used in interpreting;
3. a collection of such interpretations.
## Etymology
The Hebrew word *midrash* is derived from the root of the verb `{{transliteration|he|darash}}`{=mediawiki} (*דָּרַשׁ*), which means \"resort to, seek, seek with care, enquire, require\", forms of which appear frequently in the Bible.
The word *midrash* occurs twice in the Hebrew Bible: 2 Chronicles 13:22 \"in the *midrash* of the prophet Iddo\", and 24:27 \"in the *midrash* of the book of the kings\". Both the King James Version (KJV) and English Standard Version (ESV) translate the word as \"story\" in both instances; the Septuagint translates it as *βιβλίον* (book) in the first, as *γραφή* (writing) in the second. The meaning of the Hebrew word in these contexts is uncertain: it has been interpreted as referring to \"a body of authoritative narratives, or interpretations thereof, concerning historically important figures\" and seems to refer to a \"book\", perhaps even a \"book of interpretation\", which might make its use a foreshadowing of the technical sense that the rabbis later gave to the word.
Since the early Middle Ages the function of much of midrashic interpretation has been distinguished from that of `{{transliteration|he|peshat}}`{=mediawiki}, straight or direct interpretation aiming at the original literal meaning of a scriptural text.
## As a genre {#as_a_genre}
A definition of \"midrash\" repeatedly quoted by other scholars is that given by Gary G. Porton in 1981: \"a type of literature, oral or written, which stands in direct relationship to a fixed, canonical text, considered to be the authoritative and revealed word of God by the midrashist and his audience, and in which this canonical text is explicitly cited or clearly alluded to\".
Lieve M. Teugels, who would limit midrash to rabbinic literature, offered a definition of midrash as \"rabbinic interpretation of Scripture that bears the lemmatic form\", a definition that, unlike Porton\'s, has not been adopted by others. While some scholars agree with the limitation of the term \"midrash\" to rabbinic writings, others apply it also to certain Qumran writings, to parts of the New Testament, and of the Hebrew Bible (in particular the superscriptions of the Psalms, Deuteronomy, and Chronicles), and even modern compositions are called midrashim.
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# Midrash
## As method {#as_method}
Midrash is now viewed more as method than genre, although the rabbinic midrashim do constitute a distinct literary genre. According to the *Encyclopaedia Britannica*, \"Midrash was initially a philological method of interpreting the literal meaning of biblical texts. In time it developed into a sophisticated interpretive system that reconciled apparent biblical contradictions, established the scriptural basis of new laws, and enriched biblical content with new meaning. Midrashic creativity reached its peak in the schools of Rabbi Ishmael and Akiba, where two different hermeneutic methods were applied. The first was primarily logically oriented, making inferences based upon similarity of content and analogy. The second rested largely upon textual scrutiny, assuming that words and letters that seem superfluous teach something not openly stated in the text.\"
Many different exegetical methods are employed to derive deeper meaning from a text. This is not limited to the traditional thirteen textual tools attributed to the Tanna Rabbi Ishmael, which are used in the interpretation of `{{transliteration|he|[[halakha]]}}`{=mediawiki} (Jewish law). The presence of words or letters which are seen to be apparently superfluous, and the chronology of events, parallel narratives or what are seen as other textual \"anomalies\" are often used as a springboard for interpretation of segments of Biblical text. In many cases, a handful of lines in the Biblical narrative may become a long philosophical discussion.
Jacob Neusner distinguishes three midrash processes:
1. paraphrase: recounting the content of the biblical text in different language that may change the sense;
2. prophecy: reading the text as an account of something happening or about to happen in the interpreter\'s time;
3. parable or allegory: indicating deeper meanings of the words of the text as speaking of something other than the superficial meaning of the words or of everyday reality, as when the love of man and woman in the Song of Songs is interpreted as referring to the love between God and Israel as in Isaiah 5. Similar systems were later adopted by other religions, such as Christianity, and applied to texts such as the New Testament.
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# Midrash
## Jewish midrashic literature {#jewish_midrashic_literature}
Numerous Jewish midrashim previously preserved in manuscript form have been published in print, including those denominated as smaller or minor midrashim. Bernard H. Mehlman and Seth M. Limmer deprecate this usage, claiming that the term \"minor\" seems judgmental and \"small\" is inappropriate for midrashim, some of which are lengthy. They propose the term \"medieval midrashim instead\", since the period of their production extended from the twilight of the rabbinic age to the dawn of the Age of Enlightenment.
Generally speaking, rabbinic midrashim either focuses on religious law and practice (`{{transliteration|he|halakha}}`{=mediawiki}) or interprets the biblical narrative in relation to non-legal ethics or theology, creating homilies and parables based on the text. In the latter case, they are described as `{{transliteration|he|[[Aggadah|aggadic]]}}`{=mediawiki}.
### Halakhic midrashim {#halakhic_midrashim}
*Midrash halakha* is the name given to a group of tannaitic expositions on the first five books of the Hebrew Bible. These midrashim, written in Mishnaic Hebrew, clearly distinguish between the Biblical texts that they discuss and the rabbinic interpretation of that text. They often go beyond simple interpretation and derive or support halakha. This work is based on pre-set assumptions about the sacred and divine nature of the text and the belief in the legitimacy that accords with rabbinic interpretation.
Although this material treats the biblical texts as the authoritative word of God, it is clear that not all of the Hebrew Bible was fixed in its wording at this time, as some verses that are cited differ from the Masoretic, and accord with the Septuagint, or Samaritan Torah instead.
### Origins
With the growing canonization of the contents of the Hebrew Bible, both in terms of the books that it contained and the version of the text in them and an acceptance that new texts could not be added, there came a need to produce material that would clearly differentiate between that text and rabbinic interpretation of it. By collecting and compiling these thoughts, they could be presented in a manner that helped to refute claims that they were only human interpretations---the argument being that, by presenting the various collections of different schools of thought, each of which relied upon close study of the text, the growing difference between early biblical law and its later rabbinic interpretation could be reconciled.
### Aggadic midrashim {#aggadic_midrashim}
Midrashim that seek to explain the non-legal portions of the Hebrew Bible are sometimes referred to as `{{transliteration|he|aggadah}}`{=mediawiki} or `{{transliteration|he|Haggadah}}`{=mediawiki}.
Aggadic discussions of the non-legal parts of scripture are characterized by a much greater freedom of exposition than the halakhic midrashim (midrashim on Jewish law). Aggadic expositors availed themselves of various techniques, including sayings of prominent rabbis. These aggadic explanations could be philosophical or mystical disquisitions concerning angels, demons, paradise, Hell, the messiah, Satan, feasts and fasts, parables, legends, satirical assaults on those who practice idolatry, etc.
Some of these midrashim entail mystical teachings. The presentation is such that the midrash is a simple lesson to the uninitiated and a direct allusion, or analogy, to a mystical teaching for those educated in this area.
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# Midrash
## Classical compilations {#classical_compilations}
### Tannaitic
- **Alphabet of Rabbi Akiva**. This book is a midrash on the names of the letters of the hebrew alphabet.
- **Mekhilta**. The Mekhilta essentially functions as a commentary on the Book of Exodus. There are two versions of this midrash collection. One is *Mekhilta of Rabbi Ishmael*, the other is *Mekhilta of Rabbi Shimon ben Yochai*. The former is still studied today, while the latter was used by many medieval Jewish authorities. While the latter (bar Yohai) text was popularly circulated in manuscript form from the 11th to 16th centuries, it was lost for all practical purposes until it was rediscovered and printed in the 19th century.
- *Mekhilta of Rabbi Ishmael*. This is a halakhic commentary on Exodus, concentrating on the legal sections, from Exodus 12 to 35. It derives halakha from Biblical verses. This midrash collection was redacted into its final form around the 3rd or 4th century; its contents indicate that its sources are some of the oldest midrashim, dating back possibly to the time of Rabbi Akiva. The midrash on Exodus that was known to the Amoraim is not the same as our current mekhilta; their version was only the core of what later grew into the present form.
- *Mekhilta of Rabbi Shimon*. Based on the same core material as Mekhilta de Rabbi Ishmael, it followed a second route of commentary and editing, and eventually emerged as a distinct work. The Mekhilta of Rabbi Shimon is an exegetical midrash on Exodus 3 to 35, and is very roughly dated to near the fourth century.
- **Seder Olam Rabbah** (or simply **Seder Olam**). Traditionally attributed to the Tanna Jose ben Halafta. This work covers topics from the creation of the universe to the construction of the Second Temple in Jerusalem.
- **Sifra** on Leviticus. The Sifra work follows the tradition of Rabbi Akiva with additions from the School of Rabbi Ishmael. References in the Talmud to the Sifra are ambiguous; It is uncertain whether the texts mentioned in the Talmud are to an earlier version of our Sifra, or to the sources that the Sifra also drew upon. References to the Sifra from the time of the early medieval rabbis (and after) are to the text extant today. The core of this text developed in the mid-3rd century as a critique and commentary of the Mishnah, although subsequent additions and editing went on for some time afterwards.
- **Sifre** on Numbers and Deuteronomy, going back mainly to the schools of the same two Rabbis. This work is mainly a halakhic midrash, yet includes a long haggadic piece in sections 78--106. References in the Talmud, and in the later Geonic literature, indicate that the original core of Sifre was on the Book of Numbers, Exodus and Deuteronomy. However, transmission of the text was imperfect, and by the Middle Ages, only the commentary on Numbers and Deuteronomy remained. The core material was redacted around the middle of the 3rd century.
- **Sifri Zutta** (\"The small Sifre\"). This work is a halakhic commentary on the book of Numbers. The text of this midrash is only partially preserved in medieval works, while other portions were discovered by Solomon Schechter in his research in the famed Cairo Geniza. It seems to be older than most other midrash, coming from the early third century.
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# Midrash
## Classical compilations {#classical_compilations}
### Post-Talmudic {#post_talmudic}
- *Midrash Qohelet*, on Ecclesiastes (probably before middle of 9th century).
- *Midrash Esther*, on Esther (940 CE).
- The *Pesikta*, a compilation of homilies on special Pentateuchal and Prophetic lessons (early 8th century), in two versions:
- Pesikta Rabbati
- Pesikta de-Rav Kahana
- **Pirqe Rabbi Eliezer** (not before 8th century), a midrashic narrative of the more important events of the Pentateuch.
- **Tanchuma** or **Yelammedenu** (9th century) on the whole Pentateuch; its homilies often consist of a halakhic introduction, followed by several poems, exposition of the opening verses, and the Messianic conclusion. There are actually a number of different Midrash Tanhuma collections. The two most important are *Midrash Tanhuma Ha Nidpas*, literally the published text. This is also sometimes referred to as *Midrash Tanhuma Yelamdenu*. The other is based on a manuscript published by Solomon Buber and is usually known as *Midrash Tanhuma Buber*, much to many students\' confusion, this too is sometimes referred to as *Midrash Tanhuma Yelamdenu.* Although the first is the one most widely distributed today, when the medieval authors refer to Midrash Tanchuma, they usually mean the second.
- **Midrash Shmuel**, on the first two Books of Kings (I, II Samuel).
- **Midrash Tehillim**, on the Psalms.
- **Midrash Mishlé**, a commentary on the book of Proverbs.
- **Yalkut Shimoni**. A collection of midrash on the entire Hebrew Scriptures (Tanakh) containing both halakhic and aggadic midrash. It was compiled by Shimon ha-Darshan in the 13th century CE and is collected from over 50 other midrashic works.
- **Midrash HaGadol** (in English: the great midrash) (in Hebrew: מדרש הגדול) was written by Rabbi David Adani of Yemen (14th century). It is a compilation of aggadic midrashim on the Pentateuch taken from the two Talmuds and earlier Midrashim of Yemenite provenance.
- **Tanna Devei Eliyahu**. This work that stresses the reasons underlying the commandments, the importance of knowing Torah, prayer, and repentance, and the ethical and religious values that are learned through the Bible. It consists of two sections, Seder Eliyahu Rabbah and Seder Eliyahu Zuta. It is not a compilation but a uniform work with a single author.
- *Midrash Tadshe* (also called Baraita de-Rabbi Pinehas ben Yair):
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# Midrash
## Classical compilations {#classical_compilations}
### Midrash Rabbah {#midrash_rabbah}
- **Midrash Rabba** --- widely studied are the *Rabboth* (great commentaries), a collection of ten midrashim on different books of the Bible (namely, the five books of the Torah and the Five Megillot). Although referred to collectively as the Midrash Rabbah, they are not a cohesive work, being written by different authors in different locales in different historical eras. The ones on Exodus, Leviticus, Numbers, and Deuteronomy are chiefly made up of homilies on the Scripture sections for the Sabbath or festival, while the others are rather exegetical.
- Genesis Rabba, This text dates from the sixth century. A midrash on Genesis, it offers explanations of words and sentences and haggadic interpretations and expositions, many of which are only loosely tied to the text. It is often interlaced with maxims and parables. Its redactor drew upon earlier rabbinic sources, including the Mishnah, Tosefta, the halakhic midrashim the Targums. It apparently drew upon a version of Talmud Yerushalmi that resembles, yet was not identical to, the text that survived to present times. It was redacted sometime in the early fifth century.
- Exodus Rabbah (tenth or eleventh and twelfth century)
- Leviticus Rabbah (middle seventh century)
- Numbers Rabbah (twelfth century)
- Deuteronomy Rabbah (tenth century)
- Shir HaShirim Rabbah (Song of Songs) (probably before the middle of ninth century)
- Ruth Rabbah, (probably before the middle of ninth century)
- Lamentations Rabbah, (seventh century). *Lamentations Rabbah* has been transmitted in two versions. One edition is represented by the first printed edition (at Pesaro in 1519); the other is the Salomon Buber edition, based on manuscript J.I.4 from the Biblioteca Casanatense in Rome. This latter version (Buber\'s) is quoted by the Shulkhan Arukh, as well as medieval Jewish authorities. It was probably redacted sometime in the fifth century.
- *Ecclesiastes Rabbah*
- *Esther Rabbah*
## Contemporary Jewish midrash {#contemporary_jewish_midrash}
A wealth of literature and artwork has been created in the 20th and 21st centuries by people aspiring to create \"contemporary midrash\". Forms include poetry, prose, Bibliodrama (the acting out of Bible stories), murals, masks, and music, among others. The Institute for Contemporary Midrash was formed to facilitate these reinterpretations of sacred texts. The institute hosted several week-long intensives between 1995 and 2004, and published eight issues of *Living Text: The Journal of Contemporary Midrash* from 1997 to 2000.
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# Midrash
## Contemporary views {#contemporary_views}
According to Carol Bakhos, recent studies that use literary-critical tools to concentrate on the cultural and literary aspects of midrash have led to a rediscovery of the importance of these texts for finding insights into the rabbinic culture that created them. Midrash is increasingly seen as a literary and cultural construction, responsive to literary means of analysis.
Frank Kermode has written that midrash is an imaginative way of \"updating, enhancing, augmenting, explaining, and justifying the sacred text\". Because the Tanakh came to be seen as unintelligible or even offensive, midrash could be used as a means of rewriting it in a way that both makes it more acceptable to later ethical standards and conforms more to later notions of plausibility.
James L. Kugel, in *The Bible as It Was* (Cambridge, Massachusetts: Harvard University Press, 1997), examines a number of early Jewish and Christian texts that comment on, expand, or re-interpret passages from the first five books of the Tanakh between the third century BCE and the second century CE.
Kugel traces how and why biblical interpreters produced new meanings by the use of exegesis on ambiguities, syntactical details, unusual or awkward vocabulary, repetitions, etc. in the text. As an example, Kugel examines the different ways in which the biblical story that God\'s instructions are not to be found in heaven (Deuteronomy 30:12) has been interpreted. Baruch 3:29-4:1 states that this means that divine wisdom is not available anywhere other than in the Torah. Targum Neophyti (Deuteronomy 30:12) and b. Baba Metzia 59b claim that this text means that Torah is no longer hidden away, but has been given to humans who are then responsible for following it
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# Monomer
A **monomer** (`{{IPAc-en|ˈ|m|ɒ|n|ə|m|ər}}`{=mediawiki} `{{respell|MON|ə|mər}}`{=mediawiki}; *mono-*, \"one\" + *-mer*, \"part\") is a molecule that can react together with other monomer molecules to form a larger polymer chain or two- or three-dimensional network in a process called polymerization.`{{Quote box
| title = [[International Union of Pure and Applied Chemistry|IUPAC]] definition
| quote = '''Monomer molecule''': A molecule which can undergo polymerization, thereby contributing constitutional units to the essential structure of a [[macromolecule]].<ref>{{cite journal|title=Glossary of basic terms in polymer science (IUPAC Recommendations 1996)|journal=[[Pure and Applied Chemistry]]|year=1996|volume=68|issue=12|pages=2287–2311|doi=10.1351/pac199668122287|url=http://pac.iupac.org/publications/pac/pdf/1996/pdf/6812x2287.html|doi-access=free|last1=Jenkins|first1=A. D.|last2=Kratochvíl|first2=P.|last3=Stepto|first3=R. F. T.|last4=Suter|first4=U. W.}}</ref>
| align = right
| width = 30%
}}`{=mediawiki}
## Classification
Chemistry classifies monomers by type, and two broad classes based on the type of polymer they form.
By type:
- natural vs synthetic, e.g. glycine vs caprolactam, respectively
- polar vs nonpolar, e.g. vinyl acetate vs ethylene, respectively
- cyclic vs linear, e.g. ethylene oxide vs ethylene glycol, respectively
By type of polymer they form:
- those that participate in condensation polymerization
- those that participate in addition polymerization
Differing stoichiometry causes each class to create its respective form of polymer.
:
The polymerization of one kind of monomer gives a homopolymer. Many polymers are copolymers, meaning that they are derived from two different monomers. In the case of condensation polymerizations, the ratio of comonomers is usually 1:1. For example, the formation of many nylons requires equal amounts of a dicarboxylic acid and diamine. In the case of addition polymerizations, the comonomer content is often only a few percent. For example, small amounts of 1-octene monomer are copolymerized with ethylene to give specialized polyethylene.
## Synthetic monomers {#synthetic_monomers}
- Ethylene gas (H~2~C=CH~2~) is the monomer for polyethylene.
- Other modified ethylene derivatives include:
- tetrafluoroethylene (F~2~C=CF~2~) which leads to Teflon
- vinyl chloride (H~2~C=CHCl) which leads to PVC
- styrene (C~6~H~5~CH=CH~2~) which leads to polystyrene
- Epoxide monomers may be cross linked with themselves, or with the addition of a co-reactant, to form epoxy
- BPA is the monomer precursor for polycarbonate
- Terephthalic acid is a comonomer that, with ethylene glycol, forms polyethylene terephthalate.
- Dimethylsilicon dichloride is a monomer that, upon hydrolysis, gives polydimethylsiloxane.
- Ethyl methacrylate is an acrylic monomer that, when combined with an acrylic polymer, catalyzes and forms an acrylate plastic used to create artificial nail extensions
## Biopolymers
The term \"monomeric protein\" may also be used to describe one of the proteins making up a multiprotein complex.
## Natural monomers {#natural_monomers}
Some of the main biopolymers are listed below:
### Amino acids {#amino_acids}
For *proteins*, the monomers are amino acids. Polymerization occurs at ribosomes. Usually about 20 types of amino acid monomers are used to produce proteins. Hence proteins are not homopolymers.
### Nucleotides
For polynucleic acids (DNA/RNA), the monomers are nucleotides, each of which is made of a pentose sugar, a nitrogenous base and a phosphate group. Nucleotide monomers are found in the cell nucleus. Four types of nucleotide monomers are precursors to DNA and four different nucleotide monomers are precursors to RNA.
### Glucose and related sugars {#glucose_and_related_sugars}
For carbohydrates, the monomers are monosaccharides. The most abundant natural monomer is glucose, which is linked by glycosidic bonds into the polymers cellulose, starch, and glycogen.
### Isoprene
Isoprene is a natural monomer that polymerizes to form a natural rubber, most often *cis-*1,4-polyisoprene, but also *trans-*1,4-polymer. Synthetic rubbers are often based on butadiene, which is structurally related to isoprene
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# Missouria
The **Missouria** or **Missouri** (in their own language, **Niúachi**, also spelled **Niutachi**) are a Native American tribe that originated in the Great Lakes region of what is now the United States before European contact. The tribe belongs to the Chiwere division of the Siouan language family, together with the Ho-Chunk, Winnebago, Iowa, and Otoe.
Throughout the 17th and 18th centuries, the tribe lived in bands near the mouth of the Grand River and Missouri rivers at its confluence with the Missouri River, the mouth of the Missouri at its confluence with the Mississippi River, and in present-day Saline County, Missouri. Since Indian removal, they live primarily in Oklahoma. They are federally recognized as the Otoe-Missouria Tribe of Indians, headquartered in Red Rock, Oklahoma.
## Name
French colonists adapted a form of the Illinois language-name for the people: *Wimihsoorita*, which translates as \"One who has dugout canoes\". In their own Siouan language, the Missouri call themselves *Niúachi*, also spelled *Niutachi*, meaning \"People of the River Mouth.\" The Osage called them the *Waçux¢a,* and the Quapaw called them the Wa-ju\'-xd¢ǎ.
The state of Missouri and the Missouri River are named for the tribe.
## History
### 16th century {#th_century}
The tribe\'s oral history tells that they once lived north of the Great Lakes, where they were part of a larger tribe that included the Ho-Chunk, Iowa, and Otoe. They began migrating south in the 16th century.
### 17th century {#th_century_1}
The beginning of the 17th century, the Missouria lived near the confluence of the Grand and Missouri rivers, where they settled through the 18th century. Later, their oral history says that they split from the Otoe tribe, which belongs to the same Chiwere branch of the Siouan language, because of a love affair between the children of two tribal chiefs.
The 17th century brought hardships to the Missouria. The Sauk and Meskwaki frequently attacked them. Their society was even more disrupted by the high fatalities from epidemics of smallpox and other Eurasian infectious diseases that accompanied contact with Europeans. The French explorer Jacques Marquette contacted the tribe in 1673 and paved the way for trade with the French.
### 18th century {#th_century_2}
The Missouria migrated west of the Missouri River into Osage territory. During this time, they acquired horses and hunted bison. The French explorer Étienne de Veniard, Sieur de Bourgmont visited the people in the early 1720s. He married the daughter of a Missouria chief. They settled nearby, and Veniard created alliances with the people. He built Fort Orleans in 1723 as a trading post near present-day Brunswick, Missouri. It was occupied until 1726.
In 1730, an attack by the Sauk and Meskwaki tribes nearly destroyed the Missouria, killing hundreds. Most survivors reunited with the Otoe, while some joined the Osage and Kansa. After a smallpox outbreak in 1829, fewer than 100 Missouria survived, and they all joined the Otoe.
### 19th century {#th_century_3}
They signed treaties with the US government in 1830 and 1854 to cede their lands in Missouri. They relocated to the Otoe-Missouria reservation, created on the Big Blue River at the Kansas-Nebraska border. The US pressured the two tribes into ceding more lands in 1876 and 1881.
In 1880, the tribes split into two factions, the Coyote, who were traditionalists, and the Quakers, who were assimilationists. The Coyote settled on the Iowa Reservation in Indian Territory. The Quakers negotiated a small separate reservation in Indian Territory. By 1890, most of the Coyote band rejoined the Quakers on their reservation.
### 20th century {#th_century_4}
Under the Dawes Act, by 1907 members of the tribes were registered and allotted individual plots of land per household. The U.S. declared any excess communal land of the tribe as \"surplus\" and sold it to European-American settlers. The tribe merged with the Otoe tribe.
The Curtis Act disbanded tribal courts and governmental institutions to assimilate Native people into mainstream American society and prepare Indian Territory for statehood, but the tribe created their own court system in 1900. The Missouria were primarily farmers in the early 20th century. After oil was discovered on their lands in 1912, the U.S. government forced many of the tribe off their allotments.
### 21st century {#st_century}
Today, Missouri are part of the Otoe-Missouria Tribe of Indians. They hold the Otoe-Missouria encampment each July and host social dances and ceremonies at the Otoe-Missouria Cultural Center in Red Rock, Oklahoma.
## Population
According to the ethnographer James Mooney, the population of the tribe was about 200 families in 1702; 1000 people in 1780; 300 in 1805; 80 in 1829, when they were living with the Otoe; and 13 in 1910. Since then, their population numbers are combined with those of the Otoe
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# Memory leak
In computer science, a **memory leak** is a type of resource leak that occurs when a computer program incorrectly manages memory allocations in a way that memory which is no longer needed is not released. A memory leak may also happen when an object is stored in memory but cannot be accessed by the running code (i.e. unreachable memory). A memory leak has symptoms similar to a number of other problems and generally can only be diagnosed by a programmer with access to the program\'s source code.
A related concept is the \"space leak\", which is when a program consumes excessive memory but does eventually release it.
Because they can exhaust available system memory as an application runs, memory leaks are often the cause of or a contributing factor to software aging.
## Effects
### Minor leaks {#minor_leaks}
If a program has a memory leak and its memory usage is steadily increasing, there will not usually be an immediate symptom. In modern operating systems, normal memory used by an application is released when the application terminates. This means that a memory leak in a program that only runs for a short time may not be noticed and is rarely serious, and slow leaks can also be covered over by program restarts. Every physical system has a finite amount of memory, and if the memory leak is not contained (for example, by restarting the leaking program) it will eventually cause problems for users.
### Thrashing
Most modern consumer desktop operating systems have both main memory which is physically housed in RAM microchips, and secondary storage such as a hard drive. Memory allocation is dynamic -- each process gets as much memory as it requests. Active pages are transferred into main memory for fast access; inactive pages are pushed out to secondary storage to make room, as needed. When a single process starts consuming a large amount of memory, it usually occupies more and more of main memory, pushing other programs out to secondary storage -- usually significantly slowing performance of the system. Even if the leaking program is terminated, it may take some time for other programs to swap back into main memory, and for performance to return to normal. The resulting slowness and excessive accessing of secondary storage is known as thrashing.
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# Memory leak
## Effects
### Out-of-memory condition {#out_of_memory_condition}
If a program uses all available memory before being terminated (whether there is virtual memory or only main memory, such as on an embedded system) any attempt to allocate more memory will fail. This usually causes the program attempting to allocate the memory to terminate itself, or to generate a segmentation fault. Some programs are designed to recover from this situation (possibly by falling back on pre-reserved memory). The first program to experience the out-of-memory may or may not be the program that has the memory leak.
Some multi-tasking operating systems have special mechanisms to deal with an out-of-memory condition, such as killing processes at random (which may affect \"innocent\" processes), or killing the largest process in memory (which presumably is the one causing the problem). Some operating systems have a per-process memory limit, to prevent any one program from hogging all of the memory on the system. The disadvantage to this arrangement is that the operating system sometimes must be re-configured to allow proper operation of programs that legitimately require large amounts of memory, such as those dealing with graphics, video, or scientific calculations.
If the memory leak is in the kernel, the operating system itself will likely fail. Computers without sophisticated memory management, such as embedded systems, may also completely fail from a persistent memory leak.
## Causes of serious leaks {#causes_of_serious_leaks}
Much more serious leaks include those where:
- A program runs for a long time and consumes added memory over time, such as background tasks on servers, and especially in embedded systems which may be left running for many years
- New memory is allocated frequently for one-time tasks, such as when rendering the frames of a computer game or animated video
- A program can request memory, such as shared memory, that is not released, even when the program terminates
- Memory is very limited, such as in an embedded system or portable device, or where the program requires a very large amount of memory to begin with, leaving little margin for leaks
- A leak occurs within the operating system or memory manager
- A system device driver causes a leak
- The operating system does not automatically release memory on program termination
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# Memory leak
## Programming issues {#programming_issues}
Memory leaks are a common error in programming, especially when using languages that have no built in automatic garbage collection, such as C and C++. Typically, a memory leak occurs because dynamically allocated memory has become unreachable. The prevalence of memory leak bugs has led to the development of a number of debugging tools to detect unreachable memory. *BoundsChecker*, *Deleaker*, Memory Validator, *IBM Rational Purify*, *Valgrind*, *Parasoft Insure++*, *Dr. Memory* and *memwatch* are some of the more popular memory debuggers for C and C++ programs. \"Conservative\" garbage collection capabilities can be added to any programming language that lacks it as a built-in feature, and libraries for doing this are available for C and C++ programs. A conservative collector finds and reclaims most, but not all, unreachable memory.
Although the memory manager can recover unreachable memory, it cannot free memory that is still reachable and therefore potentially still useful. Modern memory managers therefore provide techniques for programmers to semantically mark memory with varying levels of usefulness, which correspond to varying levels of *reachability*. The memory manager does not free an object that is strongly reachable. An object is strongly reachable if it is reachable either directly by a strong reference or indirectly by a chain of strong references. (A *strong reference* is a reference that, unlike a weak reference, prevents an object from being garbage collected.) To prevent this, the developer is responsible for cleaning up references after use, typically by setting the reference to null once it is no longer needed and, if necessary, by deregistering any event listeners that maintain strong references to the object.
In general, automatic memory management is more robust and convenient for developers, as they do not need to implement freeing routines or worry about the sequence in which cleanup is performed or be concerned about whether or not an object is still referenced. It is easier for a programmer to know when a reference is no longer needed than to know when an object is no longer referenced. However, automatic memory management can impose a performance overhead, and it does not eliminate all of the programming errors that cause memory leaks.`{{facts|date=February 2025}}`{=mediawiki}
## Exploitation
Publicly accessible systems such as web servers or routers are prone to denial-of-service attacks if an attacker discovers a sequence of operations which can trigger a leak. Such a sequence is known as an exploit.
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# Memory leak
## RAII
Resource acquisition is initialization (RAII) is an approach to the problem commonly taken in C++, D, and Ada. It involves associating scoped objects with the acquired resources, and automatically releasing the resources once the objects are out of scope. Unlike garbage collection, RAII has the advantage of knowing when objects exist and when they do not. Compare the following C and C++ examples:
``` c
/* C version */
#include <stdlib.h>
void f(int n)
{
int* array = calloc(n, sizeof(int));
do_some_work(array);
free(array);
}
```
``` cpp
// C++ version
#include <vector>
void f(int n)
{
std::vector<int> array (n);
do_some_work(array);
}
```
The C version, as implemented in the example, requires explicit deallocation; the array is dynamically allocated (from the heap in most C implementations), and continues to exist until explicitly freed.
The C++ version requires no explicit deallocation; it will always occur automatically as soon as the object `array` goes out of scope, including if an exception is thrown. This avoids some of the overhead of garbage collection schemes. And because object destructors can free resources other than memory, RAII helps to prevent the leaking of input and output resources accessed through a handle, which mark-and-sweep garbage collection does not handle gracefully. These include open files, open windows, user notifications, objects in a graphics drawing library, thread synchronisation primitives such as critical sections, network connections, and connections to the Windows Registry or another database.
However, using RAII correctly is not always easy and has its own pitfalls. For instance, if one is not careful, it is possible to create dangling pointers (or references) by returning data by reference, only to have that data be deleted when its containing object goes out of scope.
D uses a combination of RAII and garbage collection, employing automatic destruction when it is clear that an object cannot be accessed outside its original scope, and garbage collection otherwise.
## External detection {#external_detection}
A \"sawtooth\" pattern of memory utilization may be an indicator of a memory leak within an application, particularly if the vertical drops coincide with reboots or restarts of that application. Care should be taken though because garbage collection points could also cause such a pattern and would show a healthy usage of the heap.
Constantly increasing memory usage is not necessarily evidence of a memory leak. Some applications will store ever increasing amounts of information in memory (e.g. as a cache). If the cache can grow so large as to cause problems, this may be a programming or design error, but is not a memory leak as the information remains nominally in use. In other cases, programs may require an unreasonably large amount of memory because the programmer has assumed memory is always sufficient for a particular task; for example, a graphics file processor might start by reading the entire contents of an image file and storing it all into memory, something that is not viable where a very large image exceeds available memory. Confirmation that excessive memory use is due to a memory leak requires access to the program code.`{{facts|date=February 2025}}`{=mediawiki}
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# Memory leak
## Examples
### Pseudocode
The following example, written in pseudocode, is intended to show how a memory leak can come about, and its effects, without needing any programming knowledge. The program in this case is part of some very simple software designed to control an elevator. This part of the program is run whenever anyone inside the elevator presses the button for a floor.
`When a button is pressed:`\
` Get some memory, which will be used to remember the floor number`\
` Put the floor number into the memory`\
` Are we already on the target floor?`\
` If so, we have nothing to do: finished`\
` Otherwise:`\
` Wait until the lift is idle`\
` Go to the required floor`\
` Release the memory we used to remember the floor number`
The memory leak would occur if the floor number requested is the same floor that the elevator is on; the condition for releasing the memory would be skipped. Each time this case occurs, more memory is leaked.
Cases like this would not usually have any immediate effects. People do not often press the button for the floor they are already on, and in any case, the elevator might have enough spare memory that this could happen hundreds or thousands of times. However, the elevator will eventually run out of memory. This could take months or years, so it might not be discovered despite thorough testing.
The consequences would be unpleasant; at the very least, the elevator would stop responding to requests to move to another floor (such as when an attempt is made to call the elevator or when someone is inside and presses the floor buttons). If other parts of the program need memory (a part assigned to open and close the door, for example), then no one would be able to enter, and if someone happens to be inside, they will become trapped (assuming the doors cannot be opened manually).
The memory leak lasts until the system is reset. For example: if the elevator\'s power were turned off or in a power outage, the program would stop running. When power was turned on again, the program would restart and all the memory would be available again, but the slow process of memory leak would restart together with the program, eventually prejudicing the correct running of the system.
The leak in the above example can be corrected by bringing the \"release\" operation outside of the conditional:
`When a button is pressed:`\
` Get some memory, which will be used to remember the floor number`\
` Put the floor number into the memory`\
` Are we already on the target floor?`\
` If not:`\
` Wait until the lift is idle`\
` Go to the required floor`\
` Release the memory we used to remember the floor number`
### C++
The following C++ program deliberately leaks memory by losing the pointer to the allocated memory.
``` c++
int main() {
int* a = new int(5);
a = nullptr;
/* The pointer in the 'a' no longer exists, and therefore cannot be freed,
but the memory is still allocated by the system.
If the program continues to create such pointers without freeing them,
it will consume memory continuously.
Therefore, a leak would occur
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# Molecular orbital
In chemistry, a **molecular orbital** is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The terms *atomic orbital* and *molecular orbital* were introduced by Robert S. Mulliken in 1932 to mean *one-electron orbital wave functions*. At an elementary level, they are used to describe the *region* of space in which a function has a significant amplitude.
In an isolated atom, the orbital electrons\' location is determined by functions called atomic orbitals. When multiple atoms combine chemically into a molecule by forming a valence chemical bond, the electrons\' locations are determined by the molecule as a whole, so the atomic orbitals combine to form molecular orbitals. The electrons from the constituent atoms occupy the molecular orbitals. Mathematically, molecular orbitals are an approximate solution to the Schrödinger equation for the electrons in the field of the molecule\'s atomic nuclei. They are usually constructed by combining atomic orbitals or hybrid orbitals from each atom of the molecule, or other molecular orbitals from groups of atoms. They can be quantitatively calculated using the Hartree--Fock or self-consistent field (SCF) methods.
Molecular orbitals are of three types: *bonding orbitals* which have an energy lower than the energy of the atomic orbitals which formed them, and thus promote the chemical bonds which hold the molecule together; *antibonding orbitals* which have an energy higher than the energy of their constituent atomic orbitals, and so oppose the bonding of the molecule, and *non-bonding orbitals* which have the same energy as their constituent atomic orbitals and thus have no effect on the bonding of the molecule.
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# Molecular orbital
## Overview
A molecular orbital (MO) can be used to represent the regions in a molecule where an electron occupying that orbital is likely to be found. Molecular orbitals are approximate solutions to the Schrödinger equation for the electrons in the electric field of the molecule\'s atomic nuclei. However calculating the orbitals directly from this equation is far too intractable a problem. Instead they are obtained from the combination of atomic orbitals, which predict the location of an electron in an atom. A molecular orbital can specify the electron configuration of a molecule: the spatial distribution and energy of one (or one pair of) electron(s). Most commonly a MO is represented as a linear combination of atomic orbitals (the LCAO-MO method), especially in qualitative or very approximate usage. They are invaluable in providing a simple model of bonding in molecules, understood through molecular orbital theory. Most present-day methods in computational chemistry begin by calculating the MOs of the system. A molecular orbital describes the behavior of one electron in the electric field generated by the nuclei and some average distribution of the other electrons. In the case of two electrons occupying the same orbital, the Pauli principle demands that they have opposite spin. Necessarily this is an approximation, and highly accurate descriptions of the molecular electronic wave function do not have orbitals (see configuration interaction).
Molecular orbitals are, in general, delocalized throughout the entire molecule. Moreover, if the molecule has symmetry elements, its nondegenerate molecular orbitals are either symmetric or antisymmetric with respect to any of these symmetries. In other words, the application of a symmetry operation **S** (e.g., a reflection, rotation, or inversion) to molecular orbital ψ results in the molecular orbital being unchanged or reversing its mathematical sign: **S**ψ = ±ψ. In planar molecules, for example, molecular orbitals are either symmetric (sigma) or antisymmetric (pi) with respect to reflection in the molecular plane. If molecules with degenerate orbital energies are also considered, a more general statement that molecular orbitals form bases for the irreducible representations of the molecule\'s symmetry group holds. The symmetry properties of molecular orbitals means that delocalization is an inherent feature of molecular orbital theory and makes it fundamentally different from (and complementary to) valence bond theory, in which bonds are viewed as localized electron pairs, with allowance for resonance to account for delocalization.
In contrast to these symmetry-adapted *canonical* molecular orbitals, localized molecular orbitals can be formed by applying certain mathematical transformations to the canonical orbitals. The advantage of this approach is that the orbitals will correspond more closely to the \"bonds\" of a molecule as depicted by a Lewis structure. As a disadvantage, the energy levels of these localized orbitals no longer have physical meaning. (The discussion in the rest of this article will focus on canonical molecular orbitals. For further discussions on localized molecular orbitals, see: natural bond orbital and sigma-pi and equivalent-orbital models.)
## Formation of molecular orbitals {#formation_of_molecular_orbitals}
Molecular orbitals arise from allowed interactions between atomic orbitals, which are allowed if the symmetries (determined from group theory) of the atomic orbitals are compatible with each other. Efficiency of atomic orbital interactions is determined from the overlap (a measure of how well two orbitals constructively interact with one another) between two atomic orbitals, which is significant if the atomic orbitals are close in energy. Finally, the number of molecular orbitals formed must be equal to the number of atomic orbitals in the atoms being combined to form the molecule.
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# Molecular orbital
## Qualitative discussion {#qualitative_discussion}
For an imprecise, but qualitatively useful, discussion of the molecular structure, the molecular orbitals can be obtained from the \"Linear combination of atomic orbitals molecular orbital method\" ansatz. Here, the molecular orbitals are expressed as linear combinations of atomic orbitals.
### Linear combinations of atomic orbitals (LCAO) {#linear_combinations_of_atomic_orbitals_lcao}
Molecular orbitals were first introduced by Friedrich Hund and Robert S. Mulliken in 1927 and 1928. The linear combination of atomic orbitals or \"LCAO\" approximation for molecular orbitals was introduced in 1929 by Sir John Lennard-Jones. His ground-breaking paper showed how to derive the electronic structure of the fluorine and oxygen molecules from quantum principles. This qualitative approach to molecular orbital theory is part of the start of modern quantum chemistry. Linear combinations of atomic orbitals (LCAO) can be used to estimate the molecular orbitals that are formed upon bonding between the molecule\'s constituent atoms. Similar to an atomic orbital, a Schrödinger equation, which describes the behavior of an electron, can be constructed for a molecular orbital as well. Linear combinations of atomic orbitals, or the sums and differences of the atomic wavefunctions, provide approximate solutions to the Hartree--Fock equations which correspond to the independent-particle approximation of the molecular Schrödinger equation. For simple diatomic molecules, the wavefunctions obtained are represented mathematically by the equations
$$\Psi = c_a \psi_a + c_b \psi_b$$
$$\Psi^* = c_a \psi_a - c_b \psi_b$$
where $\Psi$ and $\Psi^*$ are the molecular wavefunctions for the bonding and antibonding molecular orbitals, respectively, $\psi_a$ and $\psi_b$ are the atomic wavefunctions from atoms a and b, respectively, and $c_a$ and $c_b$ are adjustable coefficients. These coefficients can be positive or negative, depending on the energies and symmetries of the individual atomic orbitals. As the two atoms become closer together, their atomic orbitals overlap to produce areas of high electron density, and, as a consequence, molecular orbitals are formed between the two atoms. The atoms are held together by the electrostatic attraction between the positively charged nuclei and the negatively charged electrons occupying bonding molecular orbitals.
### Bonding, antibonding, and nonbonding MOs {#bonding_antibonding_and_nonbonding_mos}
When atomic orbitals interact, the resulting molecular orbital can be of three types: bonding, antibonding, or nonbonding.
Bonding MOs:
- Bonding interactions between atomic orbitals are constructive (in-phase) interactions.
- Bonding MOs are lower in energy than the atomic orbitals that combine to produce them.
Antibonding MOs:
- Antibonding interactions between atomic orbitals are destructive (out-of-phase) interactions, with a nodal plane where the wavefunction of the antibonding orbital is zero between the two interacting atoms
- Antibonding MOs are higher in energy than the atomic orbitals that combine to produce them.
Nonbonding MOs:
- Nonbonding MOs are the result of no interaction between atomic orbitals because of lack of compatible symmetries.
- Nonbonding MOs will have the same energy as the atomic orbitals of one of the atoms in the molecule.
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# Molecular orbital
## Qualitative discussion {#qualitative_discussion}
### Sigma and pi labels for MOs {#sigma_and_pi_labels_for_mos}
The type of interaction between atomic orbitals can be further categorized by the molecular-orbital symmetry labels σ (sigma), π (pi), δ (delta), φ (phi), γ (gamma) etc. These are the Greek letters corresponding to the atomic orbitals s, p, d, f and g respectively. The number of nodal planes containing the internuclear axis between the atoms concerned is zero for σ MOs, one for π, two for δ, three for φ and four for γ.
#### σ symmetry {#σ_symmetry}
A MO with σ symmetry results from the interaction of either two atomic s-orbitals or two atomic p~z~-orbitals. An MO will have σ-symmetry if the orbital is symmetric with respect to the axis joining the two nuclear centers, the internuclear axis. This means that rotation of the MO about the internuclear axis does not result in a phase change. A σ\* orbital, sigma antibonding orbital, also maintains the same phase when rotated about the internuclear axis. The σ\* orbital has a nodal plane that is between the nuclei and perpendicular to the internuclear axis.
#### π symmetry {#π_symmetry}
A MO with π symmetry results from the interaction of either two atomic p~x~ orbitals or p~y~ orbitals. An MO will have π symmetry if the orbital is asymmetric with respect to rotation about the internuclear axis. This means that rotation of the MO about the internuclear axis will result in a phase change. There is one nodal plane containing the internuclear axis, if real orbitals are considered.
A π\* orbital, pi antibonding orbital, will also produce a phase change when rotated about the internuclear axis. The π\* orbital also has a second nodal plane between the nuclei.
#### δ symmetry {#δ_symmetry}
A MO with δ symmetry results from the interaction of two atomic d~xy~ or d~x^2^-y^2^~ orbitals. Because these molecular orbitals involve low-energy d atomic orbitals, they are seen in transition-metal complexes. A δ bonding orbital has two nodal planes containing the internuclear axis, and a δ\* antibonding orbital also has a third nodal plane between the nuclei.
#### φ symmetry {#φ_symmetry}
Theoretical chemists have conjectured that higher-order bonds, such as phi bonds corresponding to overlap of f atomic orbitals, are possible. There is no known example of a molecule purported to contain a phi bond.
### Gerade and ungerade symmetry {#gerade_and_ungerade_symmetry}
For molecules that possess a center of inversion (centrosymmetric molecules) there are additional labels of symmetry that can be applied to molecular orbitals. Centrosymmetric molecules include:
- Homonuclear diatomics, X~2~
- Octahedral, EX~6~
- Square planar, EX~4~.
Non-centrosymmetric molecules include:
- Heteronuclear diatomics, XY
- Tetrahedral, EX~4~.
If inversion through the center of symmetry in a molecule results in the same phases for the molecular orbital, then the MO is said to have gerade (g) symmetry, from the German word for even. If inversion through the center of symmetry in a molecule results in a phase change for the molecular orbital, then the MO is said to have ungerade (u) symmetry, from the German word for odd. For a bonding MO with σ-symmetry, the orbital is σ~g~ (s\' + s\'\' is symmetric), while an antibonding MO with σ-symmetry the orbital is σ~u~, because inversion of s\' -- s\'\' is antisymmetric. For a bonding MO with π-symmetry the orbital is π~u~ because inversion through the center of symmetry for would produce a sign change (the two p atomic orbitals are in phase with each other but the two lobes have opposite signs), while an antibonding MO with π-symmetry is π~g~ because inversion through the center of symmetry for would not produce a sign change (the two p orbitals are antisymmetric by phase).
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# Molecular orbital
## Qualitative discussion {#qualitative_discussion}
### MO diagrams {#mo_diagrams}
The qualitative approach of MO analysis uses a molecular orbital diagram to visualize bonding interactions in a molecule. In this type of diagram, the molecular orbitals are represented by horizontal lines; the higher a line the higher the energy of the orbital, and degenerate orbitals are placed on the same level with a space between them. Then, the electrons to be placed in the molecular orbitals are slotted in one by one, keeping in mind the Pauli exclusion principle and Hund\'s rule of maximum multiplicity (only 2 electrons, having opposite spins, per orbital; place as many unpaired electrons on one energy level as possible before starting to pair them). For more complicated molecules, the wave mechanics approach loses utility in a qualitative understanding of bonding (although is still necessary for a quantitative approach). Some properties:
- A basis set of orbitals includes those atomic orbitals that are available for molecular orbital interactions, which may be bonding or antibonding
- The number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion or the basis set
- If the molecule has some symmetry, the degenerate atomic orbitals (with the same atomic energy) are grouped in linear combinations (called **symmetry-adapted atomic orbitals (SO)**), which belong to the representation of the symmetry group, so the wave functions that describe the group are known as **symmetry-adapted linear combinations** (**SALC**).
- The number of molecular orbitals belonging to one group representation is equal to the number of symmetry-adapted atomic orbitals belonging to this representation
- Within a particular representation, the symmetry-adapted atomic orbitals mix more if their atomic energy levels are closer.
The general procedure for constructing a molecular orbital diagram for a reasonably simple molecule can be summarized as follows:
1\. Assign a point group to the molecule.
2\. Look up the shapes of the SALCs.
3\. Arrange the SALCs of each molecular fragment in order of energy, noting first whether they stem from *s*, *p*, or *d* orbitals (and put them in the order *s* \< *p* \< *d*), and then their number of internuclear nodes.
4\. Combine SALCs of the same symmetry type from the two fragments, and from N SALCs form N molecular orbitals.
5\. Estimate the relative energies of the molecular orbitals from considerations of overlap and relative energies of the parent orbitals, and draw the levels on a molecular orbital energy level diagram (showing the origin of the orbitals).
6\. Confirm, correct, and revise this qualitative order by carrying out a molecular orbital calculation by using commercial software.
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# Molecular orbital
## Qualitative discussion {#qualitative_discussion}
### Bonding in molecular orbitals {#bonding_in_molecular_orbitals}
#### Orbital degeneracy {#orbital_degeneracy}
Molecular orbitals are said to be degenerate if they have the same energy. For example, in the homonuclear diatomic molecules of the first ten elements, the molecular orbitals derived from the p~x~ and the p~y~ atomic orbitals result in two degenerate bonding orbitals (of low energy) and two degenerate antibonding orbitals (of high energy).
#### Ionic bonds {#ionic_bonds}
In an ionic bond, oppositely charged ions are bonded by electrostatic attraction. It is possible to describe ionic bonds with molecular orbital theory by treating them as extremely polar bonds. Their bonding orbitals are very close in energy to the atomic orbitals of the anion. They are also very similar in character to the anion\'s atomic orbitals, which means the electrons are completely shifted to the anion. In computer diagrams, the orbitals are centered on the anion\'s core.
#### Bond order {#bond_order}
The bond order, or number of bonds, of a molecule can be determined by combining the number of electrons in bonding and antibonding molecular orbitals. A pair of electrons in a bonding orbital creates a bond, whereas a pair of electrons in an antibonding orbital negates a bond. For example, N~2~, with eight electrons in bonding orbitals and two electrons in antibonding orbitals, has a bond order of three, which constitutes a triple bond.
Bond strength is proportional to bond order---a greater amount of bonding produces a more stable bond---and bond length is inversely proportional to it---a stronger bond is shorter.
There are rare exceptions to the requirement of molecule having a positive bond order. Although Be~2~ has a bond order of 0 according to MO analysis, there is experimental evidence of a highly unstable Be~2~ molecule having a bond length of 245 pm and bond energy of 10 kJ/mol.
#### HOMO and LUMO {#homo_and_lumo}
The highest occupied molecular orbital and lowest unoccupied molecular orbital are often referred to as the HOMO and LUMO, respectively. The difference of the energies of the HOMO and LUMO is called the HOMO-LUMO gap. This notion is often the matter of confusion in literature and should be considered with caution. Its value is usually located between the fundamental gap (difference between ionization potential and electron affinity) and the optical gap. In addition, HOMO-LUMO gap can be related to a bulk material band gap or transport gap, which is usually much smaller than fundamental gap.
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# Molecular orbital
## Examples
### Homonuclear diatomics {#homonuclear_diatomics}
Homonuclear diatomic MOs contain equal contributions from each atomic orbital in the basis set. This is shown in the homonuclear diatomic MO diagrams for H~2~, He~2~, and Li~2~, all of which containing symmetric orbitals.
#### H~2~
As a simple MO example, consider the electrons in a hydrogen molecule, H~2~ (see molecular orbital diagram), with the two atoms labelled H\' and H\". The lowest-energy atomic orbitals, 1s\' and 1s\", do not transform according to the symmetries of the molecule. However, the following symmetry adapted atomic orbitals do:
1s\' -- 1s\" Antisymmetric combination: negated by reflection, unchanged by other operations
-------------- ---------------------------------------------------------------------------------
1s\' + 1s\" Symmetric combination: unchanged by all symmetry operations
The symmetric combination (called a bonding orbital) is lower in energy than the basis orbitals, and the antisymmetric combination (called an antibonding orbital) is higher. Because the H~2~ molecule has two electrons, they can both go in the bonding orbital, making the system lower in energy (hence more stable) than two free hydrogen atoms. This is called a covalent bond. The bond order is equal to the number of bonding electrons minus the number of antibonding electrons, divided by 2. In this example, there are 2 electrons in the bonding orbital and none in the antibonding orbital; the bond order is 1, and there is a single bond between the two hydrogen atoms.
#### He~2~
On the other hand, consider the hypothetical molecule of He~2~ with the atoms labeled He\' and He\". As with H~2~, the lowest energy atomic orbitals are the 1s\' and 1s\", and do not transform according to the symmetries of the molecule, while the symmetry adapted atomic orbitals do. The symmetric combination---the bonding orbital---is lower in energy than the basis orbitals, and the antisymmetric combination---the antibonding orbital---is higher. Unlike H~2~, with two valence electrons, He~2~ has four in its neutral ground state. Two electrons fill the lower-energy bonding orbital, σ~g~(1s), while the remaining two fill the higher-energy antibonding orbital, σ~u~\*(1s). Thus, the resulting electron density around the molecule does not support the formation of a bond between the two atoms; without a stable bond holding the atoms together, the molecule would not be expected to exist. Another way of looking at it is that there are two bonding electrons and two antibonding electrons; therefore, the bond order is 0 and no bond exists (the molecule has one bound state supported by the Van der Waals potential).
#### Li~2~
Dilithium Li~2~ is formed from the overlap of the 1s and 2s atomic orbitals (the basis set) of two Li atoms. Each Li atom contributes three electrons for bonding interactions, and the six electrons fill the three MOs of lowest energy, σ~g~(1s), σ~u~\*(1s), and σ~g~(2s). Using the equation for bond order, it is found that dilithium has a bond order of one, a single bond.
#### Noble gases {#noble_gases}
Considering a hypothetical molecule of He~2~, since the basis set of atomic orbitals is the same as in the case of H~2~, we find that both the bonding and antibonding orbitals are filled, so there is no energy advantage to the pair. HeH would have a slight energy advantage, but not as much as H~2~ + 2 He, so the molecule is very unstable and exists only briefly before decomposing into hydrogen and helium. In general, we find that atoms such as He that have full energy shells rarely bond with other atoms. Except for short-lived Van der Waals complexes, there are very few noble gas compounds known.
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# Molecular orbital
## Examples
### Heteronuclear diatomics {#heteronuclear_diatomics}
While MOs for homonuclear diatomic molecules contain equal contributions from each interacting atomic orbital, MOs for heteronuclear diatomics contain different atomic orbital contributions. Orbital interactions to produce bonding or antibonding orbitals in heteronuclear diatomics occur if there is sufficient overlap between atomic orbitals as determined by their symmetries and similarity in orbital energies.
#### HF
In hydrogen fluoride HF overlap between the H 1s and F 2s orbitals is allowed by symmetry but the difference in energy between the two atomic orbitals prevents them from interacting to create a molecular orbital. Overlap between the H 1s and F 2p~z~ orbitals is also symmetry allowed, and these two atomic orbitals have a small energy separation. Thus, they interact, leading to creation of σ and σ\* MOs and a molecule with a bond order of 1. Since HF is a non-centrosymmetric molecule, the symmetry labels g and u do not apply to its molecular orbitals.
## Quantitative approach {#quantitative_approach}
To obtain quantitative values for the molecular energy levels, one needs to have molecular orbitals that are such that the configuration interaction (CI) expansion converges fast towards the full CI limit. The most common method to obtain such functions is the Hartree--Fock method, which expresses the molecular orbitals as eigenfunctions of the Fock operator. One usually solves this problem by expanding the molecular orbitals as linear combinations of Gaussian functions centered on the atomic nuclei (see linear combination of atomic orbitals and basis set (chemistry)). The equation for the coefficients of these linear combinations is a generalized eigenvalue equation known as the Roothaan equations, which are in fact a particular representation of the Hartree--Fock equation. There are a number of programs in which quantum chemical calculations of MOs can be performed, including Spartan.
Simple accounts often suggest that experimental molecular orbital energies can be obtained by the methods of ultra-violet photoelectron spectroscopy for valence orbitals and X-ray photoelectron spectroscopy for core orbitals. This, however, is incorrect as these experiments measure the ionization energy, the difference in energy between the molecule and one of the ions resulting from the removal of one electron. Ionization energies are linked approximately to orbital energies by Koopmans\' theorem. While the agreement between these two values can be close for some molecules, it can be very poor in other cases
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# Systems Concepts
**Systems Concepts, Inc.** (now the **SC Group**), was a company co-founded by Stewart Nelson and Mike Levitt focused on making hardware products related to the DEC PDP-10 series of computers. One of its major products was the SA-10, an interface which allowed PDP-10s to be connected to disk and tape drives designed for use with the channel interfaces of IBM mainframes.
Later, Systems Concepts attempted to produce a compatible replacement for the DEC PDP-10 computers. \"Mars\" was the code name for a family of PDP-10-compatible computers built by Systems Concepts, including the initial SC-30M, the smaller SC-25, and the slower SC-20. These machines were marvels of engineering design; although not much slower than the unique Foonly F-1, they were physically smaller and consumed less power than the much slower DEC KS10 or Foonly F-2, F-3, or F-4 machines. They were also completely compatible with the DEC KL10, and ran all KL10 binaries (including the operating system) with no modifications at about 2-3 times faster than a KL10.
When DEC cancelled the Jupiter project in 1983, Systems Concepts hoped to sell their machine to customers with a software investment in PDP-10s. Their spring 1984 announcement generated excitement in the PDP-10 world. TOPS-10 was running on the Mars by the summer of 1984, and TOPS-20 by early fall. However, people at Systems Concepts were better at designing machines than at mass-producing or selling them; the company continually improved the design, but lost credibility as delivery dates continued to slip. They also overpriced; believing they were competing with the KL10 and VAX 8600 and not startups such as Sun Microsystems building workstations with comparable power at a fraction of the price. By the time SC shipped the first SC-30M to Stanford University in late 1985, most customers had already abandoned the PDP-10, usually for VMS or Unix systems. Nevertheless, a number were purchased by CompuServe, which depended on PDP-10s to run its online service and was eager to move to newer but fully compatible systems. CompuServe\'s demand for the computers outpaced Systems Concepts\' ability to produce them, so CompuServe licensed the design and built SC-designed computers itself. Other companies that purchased the SC-30 machines included Telmar, Reynolds and Reynolds, The Danish National Railway.
Peter Samson was director of marketing and program development.
SC later designed the SC-40, released in 1993, a faster follow-on to the SC-30M and SC-25. It can perform up to 8 times as fast as a DEC KL-10, and it also supports more physical memory, a larger virtual address space, and more modern input/output devices. These systems were also used at CompuServe.
In 1985, the company contracted to engineer and produce a PC-based cellular automata system for Tommaso Toffoli of MIT, called the CAM-6. The CAM-6 was a 2-card \"sandwich\" that plugged into an IBM PC slot and ran cellular automata rules at a 60 Hz update rate. Toffoli provided Forth-based software to operate the card. The production problems that plagued the company\'s computer products were demonstrated here as well, and only a few boards were produced.
Systems Concepts remained in business, having changed its name to the SC Group when it moved from California to Nevada
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# MEMS
**MEMS** (**micro-electromechanical systems**) is the technology of microscopic devices incorporating both electronic and moving parts. MEMS are made up of components between 1 and 100 micrometres in size (i.e., 0.001 to 0.1 mm), and MEMS devices generally range in size from 20 micrometres to a millimetre (i.e., 0.02 to 1.0 mm), although components arranged in arrays (e.g., digital micromirror devices) can be more than 1000 mm^2^. They usually consist of a central unit that processes data (an integrated circuit chip such as microprocessor) and several components that interact with the surroundings (such as microsensors).
Because of the large surface area to volume ratio of MEMS, forces produced by ambient electromagnetism (e.g., electrostatic charges and magnetic moments), and fluid dynamics (e.g., surface tension and viscosity) are more important design considerations than with larger scale mechanical devices. MEMS technology is distinguished from molecular nanotechnology or molecular electronics in that the latter two must also consider surface chemistry.
The potential of very small machines was appreciated before the technology existed that could make them (see, for example, Richard Feynman\'s famous 1959 lecture There\'s Plenty of Room at the Bottom). MEMS became practical once they could be fabricated using modified semiconductor device fabrication technologies, normally used to make electronics. These include molding and plating, wet etching (KOH, TMAH) and dry etching (RIE and DRIE), electrical discharge machining (EDM), and other technologies capable of manufacturing small devices.
They merge at the nanoscale into nanoelectromechanical systems (NEMS) and nanotechnology.
## History
An early example of a MEMS device is the resonant-gate transistor, an adaptation of the MOSFET, developed by Robert A. Wickstrom for Harvey C. Nathanson in 1965. Another early example is the resonistor, an electromechanical monolithic resonator patented by Raymond J. Wilfinger between 1966 and 1971. During the 1970s to early 1980s, a number of MOSFET microsensors were developed for measuring physical, chemical, biological and environmental parameters.
The term \"MEMS\" was introduced in 1986. S.C. Jacobsen (PI) and J.E. Wood (Co-PI) introduced the term \"MEMS\" by way of a proposal to DARPA (15 July 1986), titled \"Micro Electro-Mechanical Systems (MEMS)\", granted to the University of Utah. The term \"MEMS\" was presented by way of an invited talk by S.C. Jacobsen, titled \"Micro Electro-Mechanical Systems (MEMS)\", at the IEEE Micro Robots and Teleoperators Workshop, Hyannis, MA Nov. 9--11, 1987. The term \"MEMS\" was published by way of a submitted paper by J.E. Wood, S.C. Jacobsen, and K.W. Grace, titled \"SCOFSS: A Small Cantilevered Optical Fiber Servo System\", in the IEEE Proceedings Micro Robots and Teleoperators Workshop, Hyannis, MA Nov. 9--11, 1987. CMOS transistors have been manufactured on top of MEMS structures.
## Types
There are two basic types of MEMS switch technology: capacitive and ohmic. A capacitive MEMS switch is developed using a moving plate or sensing element, which changes the capacitance. Ohmic switches are controlled by electrostatically controlled cantilevers. Ohmic MEMS switches can fail from metal fatigue of the MEMS actuator (cantilever) and contact wear, since cantilevers can deform over time.
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# MEMS
## Materials
The fabrication of MEMS evolved from the process technology in semiconductor device fabrication, i.e. the basic techniques are deposition of material layers, patterning by photolithography and etching to produce the required shapes.
Silicon: Silicon is the material used to create most integrated circuits used in consumer electronics in the modern industry. The economies of scale, ready availability of inexpensive high-quality materials, and ability to incorporate electronic functionality make silicon attractive for a wide variety of MEMS applications. Silicon also has significant advantages engendered through its material properties. In single crystal form, silicon is an almost perfect Hookean material, meaning that when it is flexed there is virtually no hysteresis and hence almost no energy dissipation. As well as making for highly repeatable motion, this also makes silicon very reliable as it suffers very little fatigue and can have service lifetimes in the range of billions to trillions of cycles without breaking. Semiconductor nanostructures based on silicon are gaining increasing importance in the field of microelectronics and MEMS in particular. Silicon nanowires, fabricated through the thermal oxidation of silicon, are of further interest in electrochemical conversion and storage, including nanowire batteries and photovoltaic systems.\
Polymers: Even though the electronics industry provides an economy of scale for the silicon industry, crystalline silicon is still a complex and relatively expensive material to produce. Polymers on the other hand can be produced in huge volumes, with a great variety of material characteristics. MEMS devices can be made from polymers by processes such as injection molding, embossing or stereolithography and are especially well suited to microfluidic applications such as disposable blood testing cartridges.\
Metals: Metals can also be used to create MEMS elements. While metals do not have some of the advantages displayed by silicon in terms of mechanical properties, when used within their limitations, metals can exhibit very high degrees of reliability. Metals can be deposited by electroplating, evaporation, and sputtering processes. Commonly used metals include gold, nickel, aluminium, copper, chromium, titanium, tungsten, platinum, and silver.\
Ceramics: The nitrides of silicon, aluminium and titanium as well as silicon carbide and other ceramics are increasingly applied in MEMS fabrication due to advantageous combinations of material properties. AlN crystallizes in the wurtzite structure and thus shows pyroelectric and piezoelectric properties enabling sensors, for instance, with sensitivity to normal and shear forces. TiN, on the other hand, exhibits a high electrical conductivity and large elastic modulus, making it possible to implement electrostatic MEMS actuation schemes with ultrathin beams. Moreover, the high resistance of TiN against biocorrosion qualifies the material for applications in biogenic environments. The figure shows an electron-microscopic picture of a MEMS biosensor with a 50 nm thin bendable TiN beam above a TiN ground plate. Both can be driven as opposite electrodes of a capacitor, since the beam is fixed in electrically isolating side walls. When a fluid is suspended in the cavity its viscosity may be derived from bending the beam by electrical attraction to the ground plate and measuring the bending velocity.
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# MEMS
## Basic processes {#basic_processes}
### Deposition processes {#deposition_processes}
One of the basic building blocks in MEMS processing is the ability to deposit thin films of material with a thickness anywhere from one micrometre to about 100 micrometres. The NEMS process is the same, although the measurement of film deposition ranges from a few nanometres to one micrometre. There are two types of deposition processes, as follows.
#### Physical deposition {#physical_deposition}
Physical vapor deposition (\"PVD\") consists of a process in which a material is removed from a target, and deposited on a surface. Techniques to do this include the process of sputtering, in which an ion beam liberates atoms from a target, allowing them to move through the intervening space and deposit on the desired substrate, and evaporation, in which a material is evaporated from a target using either heat (thermal evaporation) or an electron beam (e-beam evaporation) in a vacuum system.
#### Chemical deposition {#chemical_deposition}
Chemical deposition techniques include chemical vapor deposition (CVD), in which a stream of source gas reacts on the substrate to grow the material desired. This can be further divided into categories depending on the details of the technique, for example LPCVD (low-pressure chemical vapor deposition) and PECVD (plasma-enhanced chemical vapor deposition). Oxide films can also be grown by the technique of thermal oxidation, in which the (typically silicon) wafer is exposed to oxygen and/or steam, to grow a thin surface layer of silicon dioxide.
### Patterning
Patterning is the transfer of a pattern into a material.
### Lithography
Lithography in a MEMS context is typically the transfer of a pattern into a photosensitive material by selective exposure to a radiation source such as light. A photosensitive material is a material that experiences a change in its physical properties when exposed to a radiation source. If a photosensitive material is selectively exposed to radiation (e.g. by masking some of the radiation) the pattern of the radiation on the material is transferred to the material exposed, as the properties of the exposed and unexposed regions differs.
This exposed region can then be removed or treated providing a mask for the underlying substrate. Photolithography is typically used with metal or other thin film deposition, wet and dry etching. Sometimes, photolithography is used to create structure without any kind of post etching. One example is SU8 based lens where SU8 based square blocks are generated. Then the photoresist is melted to form a semi-sphere which acts as a lens.
Electron beam lithography (often abbreviated as e-beam lithography) is the practice of scanning a beam of electrons in a patterned fashion across a surface covered with a film (called the resist), (\"exposing\" the resist) and of selectively removing either exposed or non-exposed regions of the resist (\"developing\"). The purpose, as with photolithography, is to create very small structures in the resist that can subsequently be transferred to the substrate material, often by etching. It was developed for manufacturing integrated circuits, and is also used for creating nanotechnology architectures. The primary advantage of electron beam lithography is that it is one of the ways to beat the diffraction limit of light and make features in the nanometer range. This form of maskless lithography has found wide usage in photomask-making used in photolithography, low-volume production of semiconductor components, and research & development. The key limitation of electron beam lithography is throughput, i.e., the very long time it takes to expose an entire silicon wafer or glass substrate. A long exposure time leaves the user vulnerable to beam drift or instability which may occur during the exposure. Also, the turn-around time for reworking or re-design is lengthened unnecessarily if the pattern is not being changed the second time.
It is known that focused-ion beam lithography has the capability of writing extremely fine lines (less than 50 nm line and space has been achieved) without proximity effect. However, because the writing field in ion-beam lithography is quite small, large area patterns must be created by stitching together the small fields.
Ion track technology is a deep cutting tool with a resolution limit around 8 nm applicable to radiation resistant minerals, glasses and polymers. It is capable of generating holes in thin films without any development process. Structural depth can be defined either by ion range or by material thickness. Aspect ratios up to several 10^4^ can be reached. The technique can shape and texture materials at a defined inclination angle. Random pattern, single-ion track structures and an aimed pattern consisting of individual single tracks can be generated.
X-ray lithography is a process used in the electronic industry to selectively remove parts of a thin film. It uses X-rays to transfer a geometric pattern from a mask to a light-sensitive chemical photoresist, or simply \"resist\", on the substrate. A series of chemical treatments then engraves the produced pattern into the material underneath the photoresist.
Diamond patterning is a method of forming diamond MEMS. It is achieved by the lithographic application of diamond films to a substrate such as silicon. The patterns can be formed by selective deposition through a silicon dioxide mask, or by deposition followed by micromachining or focused ion beam milling.
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# MEMS
## Basic processes {#basic_processes}
### Etching processes {#etching_processes}
There are two basic categories of etching processes: wet etching and dry etching. In the former, the material is dissolved when immersed in a chemical solution. In the latter, the material is sputtered or dissolved using reactive ions or a vapor phase etchant.
#### Wet etching {#wet_etching}
Wet chemical etching consists of the selective removal of material by dipping a substrate into a solution that dissolves it. The chemical nature of this etching process provides good selectivity, which means the etching rate of the target material is considerably higher than the mask material if selected carefully. Wet etching can be performed using either isotropic wet etchants or anisotropic wet etchants. Isotropic wet etchant etch in all directions of the crystalline silicon at approximately equal rates. Anisotropic wet etchants preferably etch along certain crystal planes at faster rates than other planes, thereby allowing more complicated 3-D microstructures to be implemented. Wet anisotropic etchants are often used in conjunction with boron etch stops wherein the surface of the silicon is heavily doped with boron resulting in a silicon material layer that is resistant to the wet etchants. This has been used in MEWS pressure sensor manufacturing for example.
Etching progresses at the same speed in all directions. Long and narrow holes in a mask will produce v-shaped grooves in the silicon. The surface of these grooves can be atomically smooth if the etch is carried out correctly, with dimensions and angles being extremely accurate.
Some single crystal materials, such as silicon, will have different etching rates depending on the crystallographic orientation of the substrate. This is known as anisotropic etching and one of the most common examples is the etching of silicon in KOH (potassium hydroxide), where Si \<111\> planes etch approximately 100 times slower than other planes (crystallographic orientations). Therefore, etching a rectangular hole in a (100)-Si wafer results in a pyramid shaped etch pit with 54.7° walls, instead of a hole with curved sidewalls as with isotropic etching.
Hydrofluoric acid is commonly used as an aqueous etchant for silicon dioxide (`{{chem|SiO|2}}`{=mediawiki}, also known as BOX for SOI), usually in 49% concentrated form, 5:1, 10:1 or 20:1 BOE (buffered oxide etchant) or BHF (Buffered HF). They were first used in medieval times for glass etching. It was used in IC fabrication for patterning the gate oxide until the process step was replaced by RIE. Hydrofluoric acid is considered one of the more dangerous acids in the cleanroom.
Electrochemical etching (ECE) for dopant-selective removal of silicon is a common method to automate and to selectively control etching. An active p--n diode junction is required, and either type of dopant can be the etch-resistant (\"etch-stop\") material. Boron is the most common etch-stop dopant. In combination with wet anisotropic etching as described above, ECE has been used successfully for controlling silicon diaphragm thickness in commercial piezoresistive silicon pressure sensors. Selectively doped regions can be created either by implantation, diffusion, or epitaxial deposition of silicon.
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# MEMS
## Basic processes {#basic_processes}
### Etching processes {#etching_processes}
#### Dry etching {#dry_etching}
Xenon difluoride (`{{chem|XeF|2}}`{=mediawiki}) is a dry vapor phase isotropic etch for silicon originally applied for MEMS in 1995 at University of California, Los Angeles. Primarily used for releasing metal and dielectric structures by undercutting silicon, `{{chem|XeF|2}}`{=mediawiki} has the advantage of a stiction-free release unlike wet etchants. Its etch selectivity to silicon is very high, allowing it to work with photoresist, `{{chem|SiO|2}}`{=mediawiki}, silicon nitride, and various metals for masking. Its reaction to silicon is \"plasmaless\", is purely chemical and spontaneous and is often operated in pulsed mode. Models of the etching action are available, and university laboratories and various commercial tools offer solutions using this approach.
Modern VLSI processes avoid wet etching, and use plasma etching instead. Plasma etchers can operate in several modes by adjusting the parameters of the plasma. Ordinary plasma etching operates between 0.1 and 5 Torr. (This unit of pressure, commonly used in vacuum engineering, equals approximately 133.3 pascals.) The plasma produces energetic free radicals, neutrally charged, that react at the surface of the wafer. Since neutral particles attack the wafer from all angles, this process is isotropic. Plasma etching can be isotropic, i.e., exhibiting a lateral undercut rate on a patterned surface approximately the same as its downward etch rate, or can be anisotropic, i.e., exhibiting a smaller lateral undercut rate than its downward etch rate. Such anisotropy is maximized in deep reactive ion etching. The use of the term anisotropy for plasma etching should not be conflated with the use of the same term when referring to orientation-dependent etching. The source gas for the plasma usually contains small molecules rich in chlorine or fluorine. For instance, carbon tetrachloride (`{{Chem2|CCl4}}`{=mediawiki}) etches silicon and aluminium, and trifluoromethane etches silicon dioxide and silicon nitride. A plasma containing oxygen is used to oxidize (\"ash\") photoresist and facilitate its removal.
Ion milling, or sputter etching, uses lower pressures, often as low as 10^−4^ Torr (10 mPa). It bombards the wafer with energetic ions of noble gases, often Ar+, which knock atoms from the substrate by transferring momentum. Because the etching is performed by ions, which approach the wafer approximately from one direction, this process is highly anisotropic. On the other hand, it tends to display poor selectivity. Reactive-ion etching (RIE) operates under conditions intermediate between sputter and plasma etching (between 10^−3^ and 10^−1^ Torr). Deep reactive-ion etching (DRIE) modifies the RIE technique to produce deep, narrow features.
In reactive-ion etching (RIE), the substrate is placed inside a reactor, and several gases are introduced. A plasma is struck in the gas mixture using an RF power source, which breaks the gas molecules into ions. The ions accelerate towards, and react with, the surface of the material being etched, forming another gaseous material. This is known as the chemical part of reactive ion etching. There is also a physical part, which is similar to the sputtering deposition process. If the ions have high enough energy, they can knock atoms out of the material to be etched without a chemical reaction. It is a very complex task to develop dry etch processes that balance chemical and physical etching, since there are many parameters to adjust. By changing the balance it is possible to influence the anisotropy of the etching, since the chemical part is isotropic and the physical part highly anisotropic the combination can form sidewalls that have shapes from rounded to vertical.
Deep reactive ion etching (DRIE) is a special subclass of RIE that is growing in popularity. In this process, etch depths of hundreds of micrometers are achieved with almost vertical sidewalls. The primary technology is based on the so-called \"Bosch process\", named after the German company Robert Bosch, which filed the original patent, where two different gas compositions alternate in the reactor. Currently, there are two variations of the DRIE. The first variation consists of three distinct steps (the original Bosch process) while the second variation only consists of two steps.
In the first variation, the etch cycle is as follows:
: \(i\) `{{chem|SF|6}}`{=mediawiki} isotropic etch;
: \(ii\) `{{chem|C|4|F|8}}`{=mediawiki} passivation;
: \(iii\) `{{chem|SF|6}}`{=mediawiki} anisotropic etch for floor cleaning.
In the 2nd variation, steps (i) and (iii) are combined.
Both variations operate similarly. The `{{chem|C|4|F|8}}`{=mediawiki} creates a polymer on the surface of the substrate, and the second gas composition (`{{chem|SF|6}}`{=mediawiki} and `{{chem|O|2}}`{=mediawiki}) etches the substrate. The polymer is immediately sputtered away by the physical part of the etching, but only on the horizontal surfaces and not the sidewalls. Since the polymer only dissolves very slowly in the chemical part of the etching, it builds up on the sidewalls and protects them from etching. As a result, etching aspect ratios of 50 to 1 can be achieved. The process can easily be used to etch completely through a silicon substrate, and etch rates are 3--6 times higher than wet etching.
After preparing a large number of MEMS devices on a silicon wafer, individual dies have to be separated, which is called die preparation in semiconductor technology. For some applications, the separation is preceded by wafer backgrinding in order to reduce the wafer thickness. Wafer dicing may then be performed either by sawing using a cooling liquid or a dry laser process called stealth dicing.
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# MEMS
## Manufacturing technologies {#manufacturing_technologies}
Bulk micromachining is the oldest paradigm of silicon-based MEMS. The whole thickness of a silicon wafer is used for building the micro-mechanical structures. Silicon is machined using various etching processes. Bulk micromachining has been essential in enabling high performance pressure sensors and accelerometers that changed the sensor industry in the 1980s and 1990s.
Surface micromachining uses layers deposited on the surface of a substrate as the structural materials, rather than using the substrate itself. Surface micromachining was created in the late 1980s to render micromachining of silicon more compatible with planar integrated circuit technology, with the goal of combining MEMS and integrated circuits on the same silicon wafer. The original surface micromachining concept was based on thin polycrystalline silicon layers patterned as movable mechanical structures and released by sacrificial etching of the underlying oxide layer. Interdigital comb electrodes were used to produce in-plane forces and to detect in-plane movement capacitively. This MEMS paradigm has enabled the manufacturing of low cost accelerometers for e.g. automotive air-bag systems and other applications where low performance and/or high g-ranges are sufficient. Analog Devices has pioneered the industrialization of surface micromachining and has realized the co-integration of MEMS and integrated circuits.
Wafer bonding involves joining two or more substrates (usually having the same diameter) to one another to form a composite structure. There are several types of wafer bonding processes that are used in microsystems fabrication including: direct or fusion wafer bonding, wherein two or more wafers are bonded together that are usually made of silicon or some other semiconductor material; anodic bonding wherein a boron-doped glass wafer is bonded to a semiconductor wafer, usually silicon; thermocompression bonding, wherein an intermediary thin-film material layer is used to facilitate wafer bonding; and eutectic bonding, wherein a thin-film layer of gold is used to bond two silicon wafers. Each of these methods have specific uses depending on the circumstances. Most wafer bonding processes rely on three basic criteria for successfully bonding: the wafers to be bonded are sufficiently flat; the wafer surfaces are sufficiently smooth; and the wafer surfaces are sufficiently clean. The most stringent criteria for wafer bonding is usually the direct fusion wafer bonding since even one or more small particulates can render the bonding unsuccessful. In comparison, wafer bonding methods that use intermediary layers are often far more forgiving.
Both bulk and surface silicon micromachining are used in the industrial production of sensors, ink-jet nozzles, and other devices. But in many cases the distinction between these two has diminished. A new etching technology, deep reactive-ion etching, has made it possible to combine good performance typical of bulk micromachining with comb structures and in-plane operation typical of surface micromachining. While it is common in surface micromachining to have structural layer thickness in the range of 2 μm, in HAR silicon micromachining the thickness can be from 10 to 100 μm. The materials commonly used in HAR silicon micromachining are thick polycrystalline silicon, known as epi-poly, and bonded silicon-on-insulator (SOI) wafers although processes for bulk silicon wafer also have been created (SCREAM). Bonding a second wafer by glass frit bonding, anodic bonding or alloy bonding is used to protect the MEMS structures. Integrated circuits are typically not combined with HAR silicon micromachining.
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# MEMS
## Applications
Some common commercial applications of MEMS include:
- Inkjet printers, which use piezoelectrics or thermal bubble ejection to deposit ink on paper.
- Accelerometers in modern cars for a large number of purposes including airbag deployment and electronic stability control.
- Inertial measurement units (IMUs):
- MEMS accelerometers.
- MEMS gyroscopes in remote controlled, or autonomous, helicopters, planes and multirotors (also known as drones), used for automatically sensing and balancing flying characteristics of roll, pitch and yaw.
- MEMS magnetic field sensor (magnetometer) may also be incorporated in such devices to provide directional heading.
- MEMS inertial navigation systems (INSs) of modern cars, airplanes, submarines and other vehicles to detect yaw, pitch, and roll; for example, the autopilot of an airplane.
- Accelerometers in consumer electronics devices such as game controllers (Nintendo Wii), personal media players / cell phones (virtually all smartphones, various HTC PDA models), augmented reality (AR) and virtual reality (VR) devices, and a number of digital cameras (various Canon Digital IXUS models). Also used in PCs to park the hard disk head when free-fall is detected, to prevent damage and data loss.
- MEMS speakers for Headphones
- MEMS barometers.
- MEMS microphones in portable devices, e.g., mobile phones, head sets and laptops. The market for smart microphones includes smartphones, wearable devices, smart home and automotive applications.
- Precision temperature-compensated resonators in real-time clocks.
- Silicon pressure sensors e.g., car tire pressure sensors, and disposable blood pressure sensors.
- Displays e.g., the digital micromirror device (DMD) chip in a projector based on DLP technology, which has a surface with several hundred thousand micromirrors or single micro-scanning-mirrors also called microscanners. The MEMS mirrors can also be used in conjunction with laser scanning to project an image.
- Optical switching technology, which is used for switching technology and alignment for data communications.
- RF switches and relays.
- Bio-MEMS applications in medical and health related technologies including lab-on-a-chip (taking advantage of microfluidics and micropumps), biosensors, chemosensors as well as embedded components of medical devices e.g. stents.
- Interferometric modulator display (IMOD) applications in consumer electronics (primarily displays for mobile devices), used to create interferometric modulation − reflective display technology as found in mirasol displays.
- Fluid acceleration, such as for micro-cooling.
- Micro-scale energy harvesting including piezoelectric, electrostatic and electromagnetic micro harvesters.
- Micromachined ultrasound transducers.
- MEMS-based loudspeakers focusing on applications such as in-ear headphones and hearing aids.
- MEMS oscillators.
- MEMS-based scanning probe microscopes including atomic force microscopes.
- LiDAR (light detection and ranging).
## Industry structure {#industry_structure}
The global market for micro-electromechanical systems, which includes products such as automobile airbag systems, display systems and inkjet cartridges totaled \$40 billion in 2006 according to Global MEMS/Microsystems Markets and Opportunities, a research report from SEMI and Yole Development and is forecasted to reach \$72 billion by 2011.
Companies with strong MEMS programs come in many sizes. Larger firms specialize in manufacturing high volume inexpensive components or packaged solutions for end markets such as automobiles, biomedical, and electronics. Smaller firms provide value in innovative solutions and absorb the expense of custom fabrication with high sales margins. Both large and small companies typically invest in R&D to explore new MEMS technology.
The market for materials and equipment used to manufacture MEMS devices topped \$1 billion worldwide in 2006. Materials demand is driven by substrates, making up over 70 percent of the market, packaging coatings and increasing use of chemical mechanical planarization (CMP). While MEMS manufacturing continues to be dominated by used semiconductor equipment, there is a migration to 200mm lines and select new tools, including etch and bonding for certain MEMS applications
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# Monoid
Monad}} `{{Algebraic structures |group}}`{=mediawiki}
In abstract algebra, a **monoid** is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being `{{math|0}}`{=mediawiki}.
Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics.
The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object.
In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing.
In theoretical computer science, the study of monoids is fundamental for automata theory (Krohn--Rhodes theory), and formal language theory (star height problem).
See semigroup for the history of the subject, and some other general properties of monoids.
## Definition
A set `{{math|''S''}}`{=mediawiki} equipped with a binary operation `{{math|''S'' × ''S'' → ''S''}}`{=mediawiki}, which we will denote `{{math|•}}`{=mediawiki}, is a **monoid** if it satisfies the following two axioms:
Associativity: For all `{{math|''a''}}`{=mediawiki}, `{{math|''b''}}`{=mediawiki} and `{{math|''c''}}`{=mediawiki} in `{{math|''S''}}`{=mediawiki}, the equation `{{math|1=(''a'' • ''b'') • ''c'' = ''a'' • (''b'' • ''c'')}}`{=mediawiki} holds.\
Identity element: There exists an element `{{math|''e''}}`{=mediawiki} in `{{math|''S''}}`{=mediawiki} such that for every element `{{math|''a''}}`{=mediawiki} in `{{math|''S''}}`{=mediawiki}, the equalities `{{math|1=''e'' • ''a'' = ''a''}}`{=mediawiki} and `{{math|1=''a'' • ''e'' = ''a''}}`{=mediawiki} hold.
In other words, a monoid is a semigroup with an identity element. It can also be thought of as a magma with associativity and identity. The identity element of a monoid is unique. For this reason the identity is regarded as a constant, i. e. `{{math|0}}`{=mediawiki}-ary (or nullary) operation. The monoid therefore is characterized by specification of the triple `{{math|(''S'', • , ''e'')}}`{=mediawiki}.
Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by juxtaposition; for example, the monoid axioms may be written `{{math|1=(''ab'')''c'' = ''a''(''bc'')}}`{=mediawiki} and `{{math|1=''ea'' = ''ae'' = ''a''}}`{=mediawiki}. This notation does not imply that it is numbers being multiplied.
A monoid in which each element has an inverse is a group.
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# Monoid
## Monoid structures {#monoid_structures}
### Submonoids
A **submonoid** of a monoid `{{math|(''M'', •)}}`{=mediawiki} is a subset `{{math|''N''}}`{=mediawiki} of `{{math|''M''}}`{=mediawiki} that is closed under the monoid operation and contains the identity element `{{math|''e''}}`{=mediawiki} of `{{math|''M''}}`{=mediawiki}. Symbolically, `{{math|''N''}}`{=mediawiki} is a submonoid of `{{math|''M''}}`{=mediawiki} if `{{math|''e'' ∈ ''N'' ⊆ ''M''}}`{=mediawiki}, and `{{math|''x'' • ''y'' ∈ ''N''}}`{=mediawiki} whenever `{{math|''x'', ''y'' ∈ ''N''}}`{=mediawiki}. In this case, `{{math|''N''}}`{=mediawiki} is a monoid under the binary operation inherited from `{{math|''M''}}`{=mediawiki}.
On the other hand, if `{{math|''N''}}`{=mediawiki} is a subset of a monoid that is closed under the monoid operation, and is a monoid for this inherited operation, then `{{math|''N''}}`{=mediawiki} is not always a submonoid, since the identity elements may differ. For example, the singleton set `{{math|{{mset|0}}}}`{=mediawiki} is closed under multiplication, and is not a submonoid of the (multiplicative) monoid of the nonnegative integers.
### Generators
A subset `{{math|''S''}}`{=mediawiki} of `{{math|''M''}}`{=mediawiki} is said to *generate* `{{math|''M''}}`{=mediawiki} if the smallest submonoid of `{{math|''M''}}`{=mediawiki} containing `{{math|''S''}}`{=mediawiki} is `{{math|''M''}}`{=mediawiki}. If there is a finite set that generates `{{math|''M''}}`{=mediawiki}, then `{{math|''M''}}`{=mediawiki} is said to be a **finitely generated monoid**.
### Commutative monoid {#commutative_monoid}
A monoid whose operation is commutative is called a **commutative monoid** (or, less commonly, an **abelian monoid**). Commutative monoids are often written additively. Any commutative monoid is endowed with its *algebraic* preordering `{{math|≤}}`{=mediawiki}, defined by `{{math|''x'' ≤ ''y''}}`{=mediawiki} if there exists `{{math|''z''}}`{=mediawiki} such that `{{math|1=''x'' + ''z'' = ''y''}}`{=mediawiki}. An *order-unit* of a commutative monoid `{{math|''M''}}`{=mediawiki} is an element `{{math|''u''}}`{=mediawiki} of `{{math|''M''}}`{=mediawiki} such that for any element `{{math|''x''}}`{=mediawiki} of `{{math|''M''}}`{=mediawiki}, there exists `{{math|''v''}}`{=mediawiki} in the set generated by `{{math|''u''}}`{=mediawiki} such that `{{math|''x'' ≤ ''v''}}`{=mediawiki}. This is often used in case `{{math|''M''}}`{=mediawiki} is the positive cone of a partially ordered abelian group `{{math|''G''}}`{=mediawiki}, in which case we say that `{{math|''u''}}`{=mediawiki} is an order-unit of `{{math|''G''}}`{=mediawiki}.
### Partially commutative monoid {#partially_commutative_monoid}
A monoid for which the operation is commutative for some, but not all elements is a trace monoid; trace monoids commonly occur in the theory of concurrent computation.
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# Monoid
## Examples
- Out of the 16 possible binary Boolean operators, four have a two-sided identity that is also commutative and associative. These four each make the set `{{math|{{mset|False, True}}}}`{=mediawiki} a commutative monoid. Under the standard definitions, AND and XNOR have the identity `{{math|True}}`{=mediawiki} while XOR and OR have the identity `{{math|False}}`{=mediawiki}. The monoids from AND and OR are also idempotent while those from XOR and XNOR are not.
- The set of natural numbers `{{math|1='''N''' = {{mset|0, 1, 2, ...}}}}`{=mediawiki} is a commutative monoid under addition (identity element `{{math|0}}`{=mediawiki}) or multiplication (identity element `{{math|1}}`{=mediawiki}). A submonoid of `{{math|'''N'''}}`{=mediawiki} under addition is called a numerical monoid.
- The set of positive integers `{{math|'''N''' ∖ {{mset|0}}}}`{=mediawiki} is a commutative monoid under multiplication (identity element `{{math|1}}`{=mediawiki}).
- Given a set `{{math|''A''}}`{=mediawiki}, the set of subsets of `{{math|''A''}}`{=mediawiki} is a commutative monoid under intersection (identity element is `{{math|''A''}}`{=mediawiki} itself).
- Given a set `{{math|''A''}}`{=mediawiki}, the set of subsets of `{{math|''A''}}`{=mediawiki} is a commutative monoid under union (identity element is the empty set).
- Generalizing the previous example, every bounded semilattice is an idempotent commutative monoid.
- In particular, any bounded lattice can be endowed with both a meet- and a join- monoid structure. The identity elements are the lattice\'s top and its bottom, respectively. Being lattices, Heyting algebras and Boolean algebras are endowed with these monoid structures.
- Every singleton set `{{math|{{mset|''x''}}}}`{=mediawiki} closed under a binary operation `{{math|•}}`{=mediawiki} forms the trivial (one-element) monoid, which is also the trivial group.
- Every group is a monoid and every abelian group a commutative monoid.
- Any semigroup `{{math|''S''}}`{=mediawiki} may be turned into a monoid simply by adjoining an element `{{math|''e''}}`{=mediawiki} not in `{{math|''S''}}`{=mediawiki} and defining `{{math|1=''e'' • ''s'' = ''s'' = ''s'' • ''e''}}`{=mediawiki} for all `{{math|''s'' ∈ ''S''}}`{=mediawiki}. This conversion of any semigroup to the monoid is done by the free functor between the category of semigroups and the category of monoids.
- Thus, an idempotent monoid (sometimes known as *find-first*) may be formed by adjoining an identity element `{{math|''e''}}`{=mediawiki} to the left zero semigroup over a set `{{math|''S''}}`{=mediawiki}. The opposite monoid (sometimes called *find-last*) is formed from the right zero semigroup over `{{math|''S''}}`{=mediawiki}.
- Adjoin an identity `{{math|''e''}}`{=mediawiki} to the left-zero semigroup with two elements `{{math|{{mset|lt, gt}}}}`{=mediawiki}. Then the resulting idempotent monoid `{{math|{{mset|lt, ''e'', gt}}}}`{=mediawiki} models the lexicographical order of a sequence given the orders of its elements, with *e* representing equality.
- The underlying set of any ring, with addition or multiplication as the operation. (By definition, a ring has a multiplicative identity `{{math|1}}`{=mediawiki}.)
- The integers, rational numbers, real numbers or complex numbers, with addition or multiplication as operation.
- The set of all `{{math|''n''}}`{=mediawiki} by `{{math|''n''}}`{=mediawiki} matrices over a given ring, with matrix addition or matrix multiplication as the operation.
- The set of all finite strings over some fixed alphabet `{{math|Σ}}`{=mediawiki} forms a monoid with string concatenation as the operation. The empty string serves as the identity element. This monoid is denoted `{{math|Σ<sup>∗</sup>}}`{=mediawiki} and is called the *free monoid* over `{{math|Σ}}`{=mediawiki}. It is not commutative if `{{math|Σ}}`{=mediawiki} has at least two elements.
- Given any monoid `{{math|''M''}}`{=mediawiki}, the *opposite monoid* `{{math|''M''<sup>op</sup>}}`{=mediawiki} has the same carrier set and identity element as `{{math|''M''}}`{=mediawiki}, and its operation is defined by `{{math|1=''x'' •<sup>op</sup> ''y'' = ''y'' • ''x''}}`{=mediawiki}. Any commutative monoid is the opposite monoid of itself.
- Given two sets `{{math|''M''}}`{=mediawiki} and `{{math|''N''}}`{=mediawiki} endowed with monoid structure (or, in general, any finite number of monoids, `{{math|''M''<sub>1</sub>, ..., ''M<sub>k</sub>''}}`{=mediawiki}), their Cartesian product `{{math|''M'' × ''N''}}`{=mediawiki}, with the binary operation and identity element defined on corresponding coordinates, called the direct product, is also a monoid (respectively, `{{math|''M''<sub>1</sub> × ⋅⋅⋅ × ''M''<sub>''k''</sub>}}`{=mediawiki}).
- Fix a monoid `{{math|''M''}}`{=mediawiki}. The set of all functions from a given set to `{{math|''M''}}`{=mediawiki} is also a monoid. The identity element is a constant function mapping any value to the identity of `{{math|''M''}}`{=mediawiki}; the associative operation is defined pointwise.
- Fix a monoid `{{math|''M''}}`{=mediawiki} with the operation `{{math|•}}`{=mediawiki} and identity element `{{math|''e''}}`{=mediawiki}, and consider its power set `{{math|''P''(''M'')}}`{=mediawiki} consisting of all subsets of `{{math|''M''}}`{=mediawiki}. A binary operation for such subsets can be defined by `{{math|1=''S'' • ''T'' = {{mset| ''s'' • ''t'' : ''s'' ∈ ''S'', ''t'' ∈ ''T'' }}}}`{=mediawiki}. This turns `{{math|''P''(''M'')}}`{=mediawiki} into a monoid with identity element `{{math|{{mset|''e''}}}}`{=mediawiki}. In the same way the power set of a group `{{math|''G''}}`{=mediawiki} is a monoid under the product of group subsets.
- Let `{{math|''S''}}`{=mediawiki} be a set. The set of all functions `{{math|''S'' → ''S''}}`{=mediawiki} forms a monoid under function composition. The identity is just the identity function. It is also called the *full transformation monoid* of `{{math|''S''}}`{=mediawiki}. If `{{math|''S''}}`{=mediawiki} is finite with `{{math|''n''}}`{=mediawiki} elements, the monoid of functions on `{{math|''S''}}`{=mediawiki} is finite with `{{math|''n''<sup>''n''</sup>}}`{=mediawiki} elements.
- Generalizing the previous example, let `{{math|''C''}}`{=mediawiki} be a category and `{{math|''X''}}`{=mediawiki} an object of `{{math|''C''}}`{=mediawiki}. The set of all endomorphisms of `{{math|''X''}}`{=mediawiki}, denoted `{{math|End<sub>''C''</sub>(''X'')}}`{=mediawiki}, forms a monoid under composition of morphisms. For more on the relationship between category theory and monoids see below.
- The set of homeomorphism classes of compact surfaces with the connected sum. Its unit element is the class of the ordinary 2-sphere. Furthermore, if `{{math|''a''}}`{=mediawiki} denotes the class of the torus, and `{{math|''b''}}`{=mediawiki} denotes the class of the projective plane, then every element `{{math|''c''}}`{=mediawiki} of the monoid has a unique expression in the form `{{math|1=''c'' = ''na'' + ''mb''}}`{=mediawiki} where `{{math|''n''}}`{=mediawiki} is a positive integer and `{{math|1=''m'' = 0, 1}}`{=mediawiki}, or `{{math|2}}`{=mediawiki}. We have `{{math|1=3''b'' = ''a'' + ''b''}}`{=mediawiki}.
- Let `{{math|{{angle bracket|{{itco|''f''}}}}}}`{=mediawiki} be a cyclic monoid of order `{{math|''n''}}`{=mediawiki}, that is, `{{math|1={{angle bracket|{{itco|''f''}}}} = {{mset|{{itco|''f''}}<sup>0</sup>, {{itco|''f''}}<sup>1</sup>, ..., {{itco|''f''}}<sup>''n''−1</sup>}}}}`{=mediawiki}. Then `{{math|1={{itco|''f''}}<sup>''n''</sup> = {{itco|''f''}}<sup>''k''</sup>}}`{=mediawiki} for some `{{math|0 ≤ ''k'' < ''n''}}`{=mediawiki}. Each such `{{math|''k''}}`{=mediawiki} gives a distinct monoid of order `{{math|''n''}}`{=mediawiki}, and every cyclic monoid is isomorphic to one of these.\
Moreover, `{{math|''f''}}`{=mediawiki} can be considered as a function on the points `{{math|{{mset|0, 1, 2, ..., ''n''−1}}}}`{=mediawiki} given by
$\begin{bmatrix}
0 & 1 & 2 & \cdots & n-2 & n-1 \\
1 & 2 & 3 & \cdots & n-1 & k\end{bmatrix}$ or, equivalently $f(i) := \begin{cases}
i+1, & \text{if } 0 \le i < n-1 \\
k, & \text{if } i = n-1.
\end{cases}$ Multiplication of elements in `{{math|{{angle bracket|{{itco|''f''}}}}}}`{=mediawiki} is then given by function composition. `{{pb}}`{=mediawiki} When `{{math|1=''k'' = 0}}`{=mediawiki} then the function `{{math|''f''}}`{=mediawiki} is a permutation of `{{math|{{mset|0, 1, 2, ..., ''n''−1}}}}`{=mediawiki}, and gives the unique cyclic group of order `{{math|''n''}}`{=mediawiki}.
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# Monoid
## Properties
The monoid axioms imply that the identity element `{{math|''e''}}`{=mediawiki} is unique: If `{{math|''e''}}`{=mediawiki} and `{{math|''f''}}`{=mediawiki} are identity elements of a monoid, then `{{math|1=''e'' = ''ef'' = ''f''}}`{=mediawiki}.
### Products and powers {#products_and_powers}
For each nonnegative integer `{{math|''n''}}`{=mediawiki}, one can define the product $p_n = \textstyle \prod_{i=1}^n a_i$ of any sequence `{{math|(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}}`{=mediawiki} of `{{math|''n''}}`{=mediawiki} elements of a monoid recursively: let `{{math|1=''p''<sub>0</sub> = ''e''}}`{=mediawiki} and let `{{math|1=''p''<sub>''m''</sub> = ''p''<sub>''m''−1</sub> • ''a''<sub>''m''</sub>}}`{=mediawiki} for `{{math|1 ≤ ''m'' ≤ ''n''}}`{=mediawiki}.
As a special case, one can define nonnegative integer powers of an element `{{math|''x''}}`{=mediawiki} of a monoid: `{{math|1=''x''<sup>0</sup> = 1}}`{=mediawiki} and `{{math|1=''x''<sup>''n''</sup> = ''x''<sup>''n''−1</sup> • ''x''}}`{=mediawiki} for `{{math|''n'' ≥ 1}}`{=mediawiki}. Then `{{math|1=''x''<sup>''m''+''n''</sup> = ''x''<sup>''m''</sup> • ''x''<sup>''n''</sup>}}`{=mediawiki} for all `{{math|''m'', ''n'' ≥ 0}}`{=mediawiki}.
### Invertible elements {#invertible_elements}
An element `{{math|''x''}}`{=mediawiki} is called invertible if there exists an element `{{math|''y''}}`{=mediawiki} such that `{{math|1=''x'' • ''y'' = ''e''}}`{=mediawiki} and `{{math|1=''y'' • ''x'' = ''e''}}`{=mediawiki}. The element `{{math|''y''}}`{=mediawiki} is called the inverse of `{{math|''x''}}`{=mediawiki}. Inverses, if they exist, are unique: if `{{math|''y''}}`{=mediawiki} and `{{math|''z''}}`{=mediawiki} are inverses of `{{math|''x''}}`{=mediawiki}, then by associativity `{{math|1=''y'' = ''ey'' = (''zx'')''y'' = ''z''(''xy'') = ''ze'' = ''z''}}`{=mediawiki}.
If `{{math|''x''}}`{=mediawiki} is invertible, say with inverse `{{math|''y''}}`{=mediawiki}, then one can define negative powers of `{{math|''x''}}`{=mediawiki} by setting `{{math|1=''x''<sup>−''n''</sup> = ''y''<sup>''n''</sup>}}`{=mediawiki} for each `{{math|''n'' ≥ 1}}`{=mediawiki}; this makes the equation `{{math|1=''x''<sup>''m''+''n''</sup> = ''x''<sup>''m''</sup> • ''x''<sup>''n''</sup>}}`{=mediawiki} hold for all `{{math|''m'', ''n'' ∈ '''Z'''}}`{=mediawiki}.
The set of all invertible elements in a monoid, together with the operation •, forms a group.
### Grothendieck group {#grothendieck_group}
Not every monoid sits inside a group. For instance, it is perfectly possible to have a monoid in which two elements `{{math|''a''}}`{=mediawiki} and `{{math|''b''}}`{=mediawiki} exist such that `{{math|1=''a'' • ''b'' = ''a''}}`{=mediawiki} holds even though `{{math|''b''}}`{=mediawiki} is not the identity element. Such a monoid cannot be embedded in a group, because in the group multiplying both sides with the inverse of `{{math|''a''}}`{=mediawiki} would get that `{{math|1=''b'' = ''e''}}`{=mediawiki}, which is not true.
A monoid `{{math|(''M'', •)}}`{=mediawiki} has the cancellation property (or is cancellative) if for all `{{math|''a''}}`{=mediawiki}, `{{math|''b''}}`{=mediawiki} and `{{math|''c''}}`{=mediawiki} in `{{math|''M''}}`{=mediawiki}, the equality `{{math|1=''a'' • ''b'' = ''a'' • ''c''}}`{=mediawiki} implies `{{math|1=''b'' = ''c''}}`{=mediawiki}, and the equality `{{math|1=''b'' • ''a'' = ''c'' • ''a''}}`{=mediawiki} implies `{{math|1=''b'' = ''c''}}`{=mediawiki}.
A commutative monoid with the cancellation property can always be embedded in a group via the *Grothendieck group construction*. That is how the additive group of the integers (a group with operation `{{math|+}}`{=mediawiki}) is constructed from the additive monoid of natural numbers (a commutative monoid with operation `{{math|+}}`{=mediawiki} and cancellation property). However, a non-commutative cancellative monoid need not be embeddable in a group.
If a monoid has the cancellation property and is *finite*, then it is in fact a group.
The right- and left-cancellative elements of a monoid each in turn form a submonoid (i.e. are closed under the operation and obviously include the identity). This means that the cancellative elements of any commutative monoid can be extended to a group.
The cancellative property in a monoid is not necessary to perform the Grothendieck construction -- commutativity is sufficient. However, if a commutative monoid does not have the cancellation property, the homomorphism of the monoid into its Grothendieck group is not injective. More precisely, if `{{math|1=''a'' • ''b'' = ''a'' • ''c''}}`{=mediawiki}, then `{{math|''b''}}`{=mediawiki} and `{{math|''c''}}`{=mediawiki} have the same image in the Grothendieck group, even if `{{math|''b'' ≠ ''c''}}`{=mediawiki}. In particular, if the monoid has an absorbing element, then its Grothendieck group is the trivial group.
### Types of monoids {#types_of_monoids}
An **inverse monoid** is a monoid where for every `{{math|''a''}}`{=mediawiki} in `{{math|''M''}}`{=mediawiki}, there exists a unique `{{math|''a''<sup>−1</sup>}}`{=mediawiki} in `{{math|''M''}}`{=mediawiki} such that `{{math|1=''a'' = ''a'' • ''a''<sup>−1</sup> • ''a''}}`{=mediawiki} and `{{math|1=''a''<sup>−1</sup> = ''a''<sup>−1</sup> • ''a'' • ''a''<sup>−1</sup>}}`{=mediawiki}. If an inverse monoid is cancellative, then it is a group.
In the opposite direction, a *zerosumfree monoid* is an additively written monoid in which `{{math|1=''a'' + ''b'' = 0}}`{=mediawiki} implies that `{{math|1=''a'' = 0}}`{=mediawiki} and `{{math|1=''b'' = 0}}`{=mediawiki}: equivalently, that no element other than zero has an additive inverse.
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# Monoid
## Acts and operator monoids {#acts_and_operator_monoids}
Let `{{math|''M''}}`{=mediawiki} be a monoid, with the binary operation denoted by `{{math|•}}`{=mediawiki} and the identity element denoted by `{{math|''e''}}`{=mediawiki}. Then a (left) **`{{math|''M''}}`{=mediawiki}-act** (or left act over `{{math|''M''}}`{=mediawiki}) is a set `{{math|''X''}}`{=mediawiki} together with an operation `{{math|⋅ : ''M'' × ''X'' → ''X''}}`{=mediawiki} which is compatible with the monoid structure as follows:
- for all `{{math|''x''}}`{=mediawiki} in `{{math|''X''}}`{=mediawiki}: `{{math|1=''e'' ⋅ ''x'' = ''x''}}`{=mediawiki};
- for all `{{math|''a''}}`{=mediawiki}, `{{math|''b''}}`{=mediawiki} in `{{math|''M''}}`{=mediawiki} and `{{math|''x''}}`{=mediawiki} in `{{math|''X''}}`{=mediawiki}: `{{math|1=''a'' ⋅ (''b'' ⋅ ''x'') = (''a'' • ''b'') ⋅ ''x''}}`{=mediawiki}.
This is the analogue in monoid theory of a (left) group action. Right `{{math|''M''}}`{=mediawiki}-acts are defined in a similar way. A monoid with an act is also known as an *operator monoid*. Important examples include transition systems of semiautomata. A transformation semigroup can be made into an operator monoid by adjoining the identity transformation.
## Monoid homomorphisms {#monoid_homomorphisms}
A homomorphism between two monoids `{{math|(''M'', ∗)}}`{=mediawiki} and `{{math|(''N'', •)}}`{=mediawiki} is a function `{{math|''f'' : ''M'' → ''N''}}`{=mediawiki} such that
- for all `{{math|''x''}}`{=mediawiki}, `{{math|''y''}}`{=mediawiki} in `{{math|''M''}}`{=mediawiki}
- ,
where `{{math|''e''<sub>''M''</sub>}}`{=mediawiki} and `{{math|''e''<sub>''N''</sub>}}`{=mediawiki} are the identities on `{{math|''M''}}`{=mediawiki} and `{{math|''N''}}`{=mediawiki} respectively. Monoid homomorphisms are sometimes simply called **monoid morphisms**.
Not every semigroup homomorphism between monoids is a monoid homomorphism, since it may not map the identity to the identity of the target monoid, even though the identity is the identity of the image of the homomorphism. For example, consider `{{math|['''Z''']<sub>''n''</sub>}}`{=mediawiki}, the set of residue classes modulo `{{math|''n''}}`{=mediawiki} equipped with multiplication. In particular, `{{math|[1]<sub>''n''</sub>}}`{=mediawiki} is the identity element. Function `{{math|''f'' : ['''Z''']<sub>3</sub> → ['''Z''']<sub>6</sub>}}`{=mediawiki} given by `{{math|[''k'']<sub>3</sub> ↦ [3''k'']<sub>6</sub>}}`{=mediawiki} is a semigroup homomorphism, since `{{math|1=[3''k'' ⋅ 3''l'']<sub>6</sub> = [9''kl'']<sub>6</sub> = [3''kl'']<sub>6</sub>}}`{=mediawiki}. However, `{{math|1=''f''([1]<sub>3</sub>) = [3]<sub>6</sub> ≠ [1]<sub>6</sub>}}`{=mediawiki}, so a monoid homomorphism is a semigroup homomorphism between monoids that maps the identity of the first monoid to the identity of the second monoid and the latter condition cannot be omitted.
In contrast, a semigroup homomorphism between groups is always a group homomorphism, as it necessarily preserves the identity (because, in the target group of the homomorphism, the identity element is the only element `{{math|''x''}}`{=mediawiki} such that `{{math|1=''x'' ⋅ ''x'' = ''x''}}`{=mediawiki}).
A bijective monoid homomorphism is called a monoid isomorphism. Two monoids are said to be isomorphic if there is a monoid isomorphism between them.
## Equational presentation {#equational_presentation}
Monoids may be given a *presentation*, much in the same way that groups can be specified by means of a group presentation. One does this by specifying a set of generators `{{math|Σ}}`{=mediawiki}, and a set of relations on the free monoid `{{math|Σ<sup>∗</sup>}}`{=mediawiki}. One does this by extending (finite) binary relations on `{{math|Σ<sup>∗</sup>}}`{=mediawiki} to monoid congruences, and then constructing the quotient monoid, as above.
Given a binary relation `{{math|''R'' ⊂ Σ<sup>∗</sup> × Σ<sup>∗</sup>}}`{=mediawiki}, one defines its symmetric closure as `{{math|''R'' ∪ ''R''<sup>−1</sup>}}`{=mediawiki}. This can be extended to a symmetric relation `{{math|''E'' ⊂ Σ<sup>∗</sup> × Σ<sup>∗</sup>}}`{=mediawiki} by defining `{{math|''x'' ~<sub>''E''</sub> ''y''}}`{=mediawiki} if and only if `{{math|1=''x'' = ''sut''}}`{=mediawiki} and `{{math|1=''y'' = ''svt''}}`{=mediawiki} for some strings `{{math|''u'', ''v'', ''s'', ''t'' ∈ Σ<sup>∗</sup>}}`{=mediawiki} with `{{math|(''u'',''v'') ∈ ''R'' ∪ ''R''<sup>−1</sup>}}`{=mediawiki}. Finally, one takes the reflexive and transitive closure of `{{math|''E''}}`{=mediawiki}, which is then a monoid congruence.
In the typical situation, the relation `{{math|''R''}}`{=mediawiki} is simply given as a set of equations, so that `{{math|1=''R'' = {{mset|1=''u''<sub>1</sub> = ''v''<sub>1</sub>, ..., ''u''<sub>''n''</sub> = ''v''<sub>''n''</sub>}}}}`{=mediawiki}. Thus, for example,
: $\langle p,q\,\vert\; pq=1\rangle$
is the equational presentation for the bicyclic monoid, and
: $\langle a,b \,\vert\; aba=baa, bba=bab\rangle$
is the plactic monoid of degree `{{math|2}}`{=mediawiki} (it has infinite order). Elements of this plactic monoid may be written as $a^ib^j(ba)^k$ for integers `{{math|''i''}}`{=mediawiki}, `{{math|''j''}}`{=mediawiki}, `{{math|''k''}}`{=mediawiki}, as the relations show that `{{math|''ba''}}`{=mediawiki} commutes with both `{{math|''a''}}`{=mediawiki} and `{{math|''b''}}`{=mediawiki}.
## Relation to category theory {#relation_to_category_theory}
Monoids can be viewed as a special class of categories. Indeed, the axioms required of a monoid operation are exactly those required of morphism composition when restricted to the set of all morphisms whose source and target is a given object. That is,
: *A monoid is, essentially, the same thing as a category with a single object.*
More precisely, given a monoid `{{math|(''M'', •)}}`{=mediawiki}, one can construct a small category with only one object and whose morphisms are the elements of `{{math|''M''}}`{=mediawiki}. The composition of morphisms is given by the monoid operation `{{math|•}}`{=mediawiki}.
Likewise, monoid homomorphisms are just functors between single object categories. So this construction gives an equivalence between the category of (small) monoids **Mon** and a full subcategory of the category of (small) categories **Cat**. Similarly, the category of groups is equivalent to another full subcategory of **Cat**.
In this sense, category theory can be thought of as an extension of the concept of a monoid. Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid.
Monoids, just like other algebraic structures, also form their own category, **Mon**, whose objects are monoids and whose morphisms are monoid homomorphisms.
There is also a notion of monoid object which is an abstract definition of what is a monoid in a category. A monoid object in **Set** is just a monoid.
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# Monoid
## Monoids in computer science {#monoids_in_computer_science}
In computer science, many abstract data types can be endowed with a monoid structure. In a common pattern, a sequence of elements of a monoid is \"folded\" or \"accumulated\" to produce a final value. For instance, many iterative algorithms need to update some kind of \"running total\" at each iteration; this pattern may be elegantly expressed by a monoid operation. Alternatively, the associativity of monoid operations ensures that the operation can be parallelized by employing a prefix sum or similar algorithm, in order to utilize multiple cores or processors efficiently.
Given a sequence of values of type `{{math|''M''}}`{=mediawiki} with identity element `{{math|''ε''}}`{=mediawiki} and associative operation `{{math|•}}`{=mediawiki}, the *fold* operation is defined as follows: $\mathrm{fold}: M^{*} \rarr M = \ell \mapsto \begin{cases} \varepsilon & \text{if } \ell = \mathrm{nil} \\
m \bullet \mathrm{fold} \, \ell' & \text{if } \ell = \mathrm{cons} \, m \, \ell' \end{cases}$
In addition, any data structure can be \'folded\' in a similar way, given a serialization of its elements. For instance, the result of \"folding\" a binary tree might differ depending on pre-order vs. post-order tree traversal.
## MapReduce
An application of monoids in computer science is the so-called MapReduce programming model (see [Encoding Map-Reduce As A Monoid With Left Folding](https://erikerlandson.github.io/blog/2016/09/05/expressing-map-reduce-as-a-left-folding-monoid/)). MapReduce, in computing, consists of two or three operations. Given a dataset, \"Map\" consists of mapping arbitrary data to elements of a specific monoid. \"Reduce\" consists of folding those elements, so that in the end we produce just one element.
For example, if we have a multiset, in a program it is represented as a map from elements to their numbers. Elements are called keys in this case. The number of distinct keys may be too big, and in this case, the multiset is being sharded. To finalize reduction properly, the \"Shuffling\" stage regroups the data among the nodes. If we do not need this step, the whole Map/Reduce consists of mapping and reducing; both operations are parallelizable, the former due to its element-wise nature, the latter due to associativity of the monoid.
## Complete monoids {#complete_monoids}
A **complete monoid** is a commutative monoid equipped with an infinitary sum operation $\Sigma_I$ for any index set `{{math|''I''}}`{=mediawiki} such that $\sum_{i \in \emptyset}{m_i} = 0;\quad \sum_{i \in \{j\}}{m_i} = m_j;\quad \sum_{i \in \{j, k\}}{m_i} = m_j+m_k \quad \text{ for } j\neq k$ and $\sum_{j \in J} {\sum_{i \in I_j}{m_i}} = \sum_{i \in I} m_i \quad \text{ if } \bigcup_{j\in J} I_j=I \text{ and } I_j \cap I_{j'} = \emptyset \quad \text{ for } j\neq j'$.
An **ordered commutative monoid** is a commutative monoid `{{math|''M''}}`{=mediawiki} together with a partial ordering `{{math|≤}}`{=mediawiki} such that `{{math|''a'' ≥ 0}}`{=mediawiki} for every `{{math|''a'' ∈ ''M''}}`{=mediawiki}, and `{{math|''a'' ≤ ''b''}}`{=mediawiki} implies `{{math|''a'' + ''c'' ≤ ''b'' + ''c''}}`{=mediawiki} for all `{{math|''a'', ''b'', ''c'' ∈ ''M''}}`{=mediawiki}.
A **continuous monoid** is an ordered commutative monoid `{{math|(''M'', ≤)}}`{=mediawiki} in which every directed subset has a least upper bound, and these least upper bounds are compatible with the monoid operation: $a + \sup S = \sup(a + S)$ for every `{{math|''a'' ∈ ''M''}}`{=mediawiki} and directed subset `{{math|''S''}}`{=mediawiki} of `{{math|''M''}}`{=mediawiki}.
If `{{math|(''M'', ≤)}}`{=mediawiki} is a continuous monoid, then for any index set `{{math|''I''}}`{=mediawiki} and collection of elements `{{math|(''a''{{sub|''i''}}){{sub|''i''∈''I''}}}}`{=mediawiki}, one can define $\sum_I a_i = \sup_{\text{finite } E \subset I} \; \sum_E a_i,$ and `{{math|''M''}}`{=mediawiki} together with this infinitary sum operation is a complete monoid
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# May 23
| 3 |
May 23
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# Michael Ventris
**Michael George Francis Ventris**, `{{post-nominals|country=GBR|size=100%|OBE}}`{=mediawiki} (`{{IPAc-en|ˈ|v|ɛ|n|t|r|ɪ|s}}`{=mediawiki}; 12 July 1922 -- 6 September 1956) was an English architect, classicist and philologist who deciphered Linear B, the ancient Mycenaean Greek script. A student of languages, Ventris had pursued decipherment as a personal vocation since his adolescence. After creating a new field of study, Ventris died in a car crash a few weeks before the publication of *Documents in Mycenaean Greek*, written with John Chadwick.
## Early life and education {#early_life_and_education}
Ventris was born into a traditional army family. His grandfather, Francis Ventris, was a major-general and Commander of British Forces in China. His father, Edward Francis Vereker Ventris, was a lieutenant-colonel in the Indian Army, who retired early due to ill health. Edward Ventris married Anna Dorothea (Dora) Janasz, who was from a wealthy Jewish and Polish paternal background. Michael Ventris was their only child.
The family moved to Switzerland for eight years, seeking a healthy environment for Colonel Ventris. Young Michael started school in Gstaad, where classes were taught in French and German. He soon mastered the Swiss German dialect. A few weeks in Sweden after the Second World War enabled him to become competent in Swedish. His mother often spoke Polish, and he was fluent in it by the age of eight. In 1931, the Ventris family returned home. From 1931 to 1935, Ventris was sent to Bickley Hall School in Bromley, Kent. His parents divorced in 1935. At this time, he secured a scholarship to Stowe School. At Stowe he learned some Latin and Greek. He did not do outstanding work there -- by then he was spending most of his spare time learning as much as he could about Linear B, some of his study time being spent under the covers at night with a torch.
When he was not boarding at school, Ventris lived with his mother, before 1935 in coastal hotels, and then in the avant garde Berthold Lubetkin\'s Highpoint modernist apartments in Highgate, north London. His mother\'s acquaintances, who frequented the house, included many sculptors, painters, and writers of the day. The flat was furnished with the works of Marcel Breuer. The money for her artistic patronage came from Polish estates.
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# Michael Ventris
## Young adult {#young_adult}
Ventris\'s father died in 1938, and his mother, Dora, became administrator of the estate. With the German invasion of Poland in 1939, Dora lost her private income, and in 1940 her father died. Ventris lost his mother to clinical depression and an overdose of barbiturates. He never spoke of her, assuming instead an ebullient and energetic manner in whatever he decided to do, a trait which won him numerous friends.
A friend of the family, Russian sculptor Naum Gabo, took Ventris under his wing. Ventris later said that Gabo was the most family he had ever had. It may have been at Gabo\'s house that he began the study of Russian.
He decided on architecture as a career, and enrolled in the Architectural Association School of Architecture. There he met his wife-to-be Lois Knox-Niven, known as Betty, daughter of pilot Lois Butler and stepdaughter of Alan Samuel Butler, chairman of the De Havilland Aircraft Company. A fellow architecture student, her social background was similar to Ventris\'s: her family was well-to-do, she had travelled in Europe, and she was interested in architecture. She was also popular and very beautiful.
thumb\|upright=1.4\|left\|Halifax bomber in flight, 1942 Ventris did not complete his architecture studies, being conscripted in 1942. He chose the Royal Air Force (RAF). His preference was for navigator rather than pilot, and he completed the extensive training in the UK and Canada, to qualify early in 1944 and be commissioned. While training, he studied Russian intensively for several weeks, the purpose of which is not clear. He took part in the bombing of Germany, as aircrew on the Handley Page Halifax with No. 76 Squadron RAF, initially at RAF Breighton and then at RAF Holme-on-Spalding Moor, both in East Yorkshire. After the conclusion of the war, he served out the rest of his term on the ground in Germany, for which he was chosen because of his knowledge of Russian. His duties are unclear. His friends assumed he was on intelligence duties, interpreting his denials as part of a legal gag. No evidence of such assignments has emerged in the decades since. There is also no evidence that he was ever part of any code-breaking unit, as was Chadwick, even though the public has readily believed this explanation of his genius and success with Linear B.
## Architect and palaeographer {#architect_and_palaeographer}
After the war, he worked briefly in Sweden, learning enough Swedish to communicate with scholars. Then he came home to complete his architectural education with honours in 1948 and settled down with Lois working as an architect. He designed schools for the Ministry of Education. He and his wife personally designed their family home, 19 North End, Hampstead. Ventris and his wife had two children: a son, Nikki (1942--1984), and a daughter, Tessa (born 1946).
Ventris continued with his efforts on Linear B, discovering in 1952 that it was an archaic form of Greek.
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